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href="https://de.wikipedia.org/wiki/Va%C5%A1ek_Chv%C3%A1tal" title="Vašek Chvátal – German" lang="de" hreflang="de" data-title="Vašek Chvátal" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%92%CE%AC%CF%84%CF%83%CE%BB%CE%B1%CE%B2_%CE%A7%CE%B2%CE%AC%CF%84%CE%B1%CE%BB" title="Βάτσλαβ Χβάταλ – Greek" lang="el" hreflang="el" data-title="Βάτσλαβ Χβάταλ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/V%C3%A1clav_Chv%C3%A1tal" title="Václav Chvátal – Spanish" lang="es" hreflang="es" data-title="Václav Chvátal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%88%D8%A7%D8%AA%D8%B3%D9%84%D8%A7%D9%81_%D8%AE%D9%88%D8%A7%D8%AA%D8%A7%D9%84" title="واتسلاف خواتال – Persian" lang="fa" hreflang="fa" data-title="واتسلاف خواتال" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/V%C3%A1clav_Chv%C3%A1tal" title="Václav Chvátal – French" lang="fr" hreflang="fr" data-title="Václav Chvátal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/V%C3%A1clav_Chv%C3%A1tal" title="Václav Chvátal – Haitian Creole" lang="ht" hreflang="ht" data-title="Václav Chvátal" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%DA%A4%D8%A7%D9%83%D9%84%D8%A7%DA%A4_%D8%AA%D8%B4%DA%A4%D8%A7%D8%AA%D8%A7%D9%84" title="ڤاكلاڤ تشڤاتال – Egyptian Arabic" lang="arz" hreflang="arz" data-title="ڤاكلاڤ تشڤاتال" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/V%C3%A1clav_Chv%C3%A1tal" title="Václav Chvátal – Portuguese" lang="pt" hreflang="pt" data-title="Václav Chvátal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Czech-Canadian mathematician</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Václav Chvátal</div></th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Va%C5%A1ek_Chv%C3%A1tal_in_2020.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Va%C5%A1ek_Chv%C3%A1tal_in_2020.jpg/160px-Va%C5%A1ek_Chv%C3%A1tal_in_2020.jpg" decoding="async" width="160" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Va%C5%A1ek_Chv%C3%A1tal_in_2020.jpg/240px-Va%C5%A1ek_Chv%C3%A1tal_in_2020.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Va%C5%A1ek_Chv%C3%A1tal_in_2020.jpg/320px-Va%C5%A1ek_Chv%C3%A1tal_in_2020.jpg 2x" data-file-width="707" data-file-height="981" /></a></span><div class="infobox-caption">Václav Chvátal (2020)</div></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><span style="display:none"> (<span class="bday">1946-07-20</span>) </span>20 July 1946<span class="noprint ForceAgeToShow"> (age 78)</span><br /><div style="display:inline" class="birthplace"><a href="/wiki/Prague" title="Prague">Prague</a></div></td></tr><tr><th scope="row" class="infobox-label">Nationality</th><td class="infobox-data category"><a href="/wiki/Canadians" title="Canadians">Canadian</a>, <a href="/wiki/Czech_Republic" title="Czech Republic">Czech</a></td></tr><tr><th scope="row" class="infobox-label">Alma mater</th><td class="infobox-data"><a href="/wiki/University_of_Waterloo" title="University of Waterloo">University of Waterloo</a><br /><a href="/wiki/Charles_University" title="Charles University">Charles University</a></td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data"><a href="/wiki/Chv%C3%A1tal_graph" title="Chvátal graph">Chvátal graph</a><br /><a href="/wiki/Chv%C3%A1tal%E2%80%93Sankoff_constants" title="Chvátal–Sankoff constants">Chvátal–Sankoff constants</a><br /><a href="/wiki/Hamiltonian_path#Bondy–Chvátal_theorem" title="Hamiltonian path">Bondy–Chvátal theorem</a><br /><a href="/wiki/Crossing_number_inequality" title="Crossing number inequality">Crossing number inequality</a><br /><a href="/wiki/Graph_toughness" title="Graph toughness">Graph toughness</a></td></tr><tr><th scope="row" class="infobox-label">Awards</th><td class="infobox-data"><a href="/w/index.php?title=Beale%E2%80%93Orchard-Hays_Prize&action=edit&redlink=1" class="new" title="Beale–Orchard-Hays Prize (page does not exist)">Beale–Orchard-Hays Prize</a> (2000) <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><br /> Docteur Honoris Causa, <a href="/w/index.php?title=Universit%C3%A9_de_la_M%C3%A9diterrann%C3%A9&action=edit&redlink=1" class="new" title="Université de la Méditerranné (page does not exist)">Université de la Méditerranné</a> (2003) <br /><a href="/wiki/Frederick_W._Lanchester_Prize" title="Frederick W. Lanchester Prize">Frederick W. Lanchester Prize</a> (2007) <sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><br /><a href="/wiki/John_von_Neumann_Theory_Prize" title="John von Neumann Theory Prize">John von Neumann Theory Prize</a> (2015) <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a>, <a href="/wiki/Computer_Science" class="mw-redirect" title="Computer Science">Computer Science</a>, <a href="/wiki/Operations_Research" class="mw-redirect" title="Operations Research">Operations Research</a></td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data"><a href="/wiki/Concordia_University" title="Concordia University">Concordia University</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisor</a></th><td class="infobox-data"><a href="/wiki/Crispin_Nash-Williams" title="Crispin Nash-Williams">Crispin Nash-Williams</a></td></tr><tr><th scope="row" class="infobox-label">Doctoral students</th><td class="infobox-data"><a href="/wiki/David_Avis" title="David Avis">David Avis</a> (Stanford 1977)<br /><a href="/wiki/Bruce_Reed_(mathematician)" title="Bruce Reed (mathematician)">Bruce Reed</a> (McGill 1986)</td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Václav (Vašek) Chvátal</b> (<style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small">Czech:</span> <span class="IPA nowrap" lang="cs-Latn-fonipa"><a href="/wiki/Help:IPA/Czech" title="Help:IPA/Czech">[ˈvaːtslaf<span class="wrap"> </span>ˈxvaːtal]</a></span>) is a Professor Emeritus in the Department of Computer Science and Software Engineering at <a href="/wiki/Concordia_University_(Montreal)" class="mw-redirect" title="Concordia University (Montreal)">Concordia University</a> in <a href="/wiki/Montreal,_Quebec" class="mw-redirect" title="Montreal, Quebec">Montreal, Quebec</a>, Canada, and a visiting professor at <a href="/wiki/Charles_University" title="Charles University">Charles University</a> in <a href="/wiki/Prague" title="Prague">Prague</a>. He has published extensively on topics in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, and <a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Biography">Biography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=V%C3%A1clav_Chv%C3%A1tal&action=edit&section=1" title="Edit section: Biography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Chvátal was born in 1946 in Prague and educated in mathematics at <a href="/wiki/Charles_University" title="Charles University">Charles University</a> in Prague, where he studied under the supervision of <a href="/wiki/Zden%C4%9Bk_Hedrl%C3%ADn" title="Zdeněk Hedrlín">Zdeněk Hedrlín</a>.<sup id="cite_ref-abcr_4-0" class="reference"><a href="#cite_note-abcr-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> He fled <a href="/wiki/Czechoslovakia" title="Czechoslovakia">Czechoslovakia</a> in 1968, three days after the <a href="/wiki/Prague_Spring" title="Prague Spring">Soviet invasion</a>,<sup id="cite_ref-travprof_5-0" class="reference"><a href="#cite_note-travprof-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and completed his Ph.D. in Mathematics at the <a href="/wiki/University_of_Waterloo" title="University of Waterloo">University of Waterloo</a>, under the supervision of <a href="/wiki/Crispin_St._J._A._Nash-Williams" class="mw-redirect" title="Crispin St. J. A. Nash-Williams">Crispin St. J. A. Nash-Williams</a>, in the fall of 1970.<sup id="cite_ref-abcr_4-1" class="reference"><a href="#cite_note-abcr-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Subsequently, he took positions at <a href="/wiki/McGill_University" title="McGill University">McGill University</a> (1971 and 1978–1986), <a href="/wiki/Stanford_University" title="Stanford University">Stanford University</a> (1972 and 1974–1977), the <a href="/wiki/Universit%C3%A9_de_Montr%C3%A9al" title="Université de Montréal">Université de Montréal</a> (1972–1974 and 1977–1978), and <a href="/wiki/Rutgers_University" title="Rutgers University">Rutgers University</a> (1986-2004) before returning to Montreal for the <a href="/wiki/Canada_Research_Chair" title="Canada Research Chair">Canada Research Chair</a> in Combinatorial Optimization <sup id="cite_ref-canchair_7-0" class="reference"><a href="#cite_note-canchair-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-travprof_5-1" class="reference"><a href="#cite_note-travprof-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> at <a href="/wiki/Concordia_University_(Montreal)" class="mw-redirect" title="Concordia University (Montreal)">Concordia</a> (2004-2011) and the <a href="/wiki/Canada_Research_Chair" title="Canada Research Chair">Canada Research Chair</a> in Discrete Mathematics (2011-2014) till his retirement. </p> <div class="mw-heading mw-heading2"><h2 id="Research">Research</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=V%C3%A1clav_Chv%C3%A1tal&action=edit&section=2" title="Edit section: Research"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Chv%C3%A1tal_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Chv%C3%A1tal_graph.svg/220px-Chv%C3%A1tal_graph.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Chv%C3%A1tal_graph.svg/330px-Chv%C3%A1tal_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Chv%C3%A1tal_graph.svg/440px-Chv%C3%A1tal_graph.svg.png 2x" data-file-width="504" data-file-height="504" /></a><figcaption>The <a href="/wiki/Chv%C3%A1tal_graph" title="Chvátal graph">Chvátal graph</a></figcaption></figure> <p>Chvátal first learned of graph theory in 1964, on finding a book by <a href="/wiki/Claude_Berge" title="Claude Berge">Claude Berge</a> in a <a href="/wiki/Plze%C5%88" title="Plzeň">Pilsen</a> bookstore <sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and much of his research involves graph theory: </p> <ul><li>His first mathematical publication, at the age of 19, concerned <a href="/wiki/Directed_graph" title="Directed graph">directed graphs</a> that cannot be mapped to themselves by any nontrivial <a href="/wiki/Graph_homomorphism" title="Graph homomorphism">graph homomorphism</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>Another graph-theoretic result of Chvátal was the 1970 construction of the smallest possible <a href="/wiki/Triangle-free_graph" title="Triangle-free graph">triangle-free graph</a> that is both 4-<a href="/wiki/Chromatic_number" class="mw-redirect" title="Chromatic number">chromatic</a> and 4-<a href="/wiki/Regular_graph" title="Regular graph">regular</a>, now known as the <a href="/wiki/Chv%C3%A1tal_graph" title="Chvátal graph">Chvátal graph</a>.<sup id="cite_ref-abcr_4-2" class="reference"><a href="#cite_note-abcr-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>A 1972 paper <sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> relating Hamiltonian cycles to connectivity and <a href="/wiki/Maximum_independent_set" class="mw-redirect" title="Maximum independent set">maximum independent set</a> size of a graph, earned Chvátal his <a href="/wiki/Erd%C5%91s_number" title="Erdős number">Erdős number</a> of 1. Specifically, if there exists an <i>s</i> such that a given graph is <i>s</i>-<a href="/wiki/K-vertex-connected_graph" title="K-vertex-connected graph">vertex-connected</a> and has no (<i>s</i> + 1)-vertex independent set, the graph must be Hamiltonian. Avis et al.<sup id="cite_ref-abcr_4-3" class="reference"><a href="#cite_note-abcr-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> tell the story of Chvátal and <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Erdős</a> working out this result over the course of a long road trip, and later thanking Louise Guy "for her steady driving."</li> <li>In a 1973 paper,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Chvátal introduced the concept of <a href="/wiki/Graph_toughness" title="Graph toughness">graph toughness</a>, a measure of <a href="/wiki/Graph_connectivity" class="mw-redirect" title="Graph connectivity">graph connectivity</a> that is closely connected to the existence of <a href="/wiki/Hamiltonian_cycle" class="mw-redirect" title="Hamiltonian cycle">Hamiltonian cycles</a>. A graph is <i>t</i>-tough if, for every <i>k</i> greater than 1, the removal of fewer than <i>tk</i> vertices leaves fewer than <i>k</i> connected components in the remaining subgraph. For instance, in a graph with a Hamiltonian cycle, the removal of any nonempty set of vertices partitions the cycle into at most as many pieces as the number of removed vertices, so Hamiltonian graphs are 1-tough. Chvátal conjectured that 3/2-tough graphs, and later that 2-tough graphs, are always Hamiltonian; despite later researchers finding counterexamples to these conjectures, it still remains open whether some constant bound on the graph toughness is enough to guarantee Hamiltonicity.<sup id="cite_ref-lesniak_13-0" class="reference"><a href="#cite_note-lesniak-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul> <p>Some of Chvátal's work concerns families of sets, or equivalently <a href="/wiki/Hypergraph" title="Hypergraph">hypergraphs</a>, a subject already occurring in his Ph.D. thesis, where he also studied <a href="/wiki/Ramsey_theory" title="Ramsey theory">Ramsey theory</a>. </p> <ul><li>In a 1972 conjecture that Erdős called "surprising" and "beautiful",<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and that remains open (with a $10 prize offered by Chvátal for its solution) <sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> he suggested that, in any family of sets closed under the operation of taking <a href="/wiki/Subset" title="Subset">subsets</a>, the largest pairwise-intersecting subfamily may always be found by choosing an element of one of the sets and keeping all sets containing that element.</li> <li>In 1979,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> he studied a weighted version of the <a href="/wiki/Set_cover_problem" title="Set cover problem">set cover problem</a>, and proved that a <a href="/wiki/Greedy_algorithm" title="Greedy algorithm">greedy algorithm</a> provides good <a href="/wiki/Approximation_algorithm" title="Approximation algorithm">approximations</a> to the optimal solution, generalizing previous unweighted results by <a href="/wiki/David_S._Johnson" title="David S. Johnson">David S. Johnson</a> (J. Comp. Sys. Sci. 1974) and <a href="/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">László Lovász</a> (Discrete Math. 1975).</li></ul> <p>Chvátal first became interested in <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a> through the influence of <a href="/wiki/Jack_Edmonds" title="Jack Edmonds">Jack Edmonds</a> while Chvátal was a student at Waterloo.<sup id="cite_ref-abcr_4-4" class="reference"><a href="#cite_note-abcr-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> He quickly recognized the importance of <a href="/wiki/Cutting-plane_method" title="Cutting-plane method">cutting planes</a> for attacking combinatorial optimization problems such as computing <a href="/wiki/Maximum_independent_set" class="mw-redirect" title="Maximum independent set">maximum independent sets</a> and, in particular, introduced the notion of a cutting-plane proof.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> At Stanford in the 1970s, he began writing his popular textbook, <i>Linear Programming</i>, which was published in 1983.<sup id="cite_ref-abcr_4-5" class="reference"><a href="#cite_note-abcr-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Cutting planes lie at the heart of the <a href="/wiki/Branch_and_cut" title="Branch and cut">branch and cut</a> method used by efficient solvers for the <a href="/wiki/Traveling_salesman_problem" class="mw-redirect" title="Traveling salesman problem">traveling salesman problem</a>. Between 1988 and 2005, the team of <a href="/wiki/David_Applegate" title="David Applegate">David L. Applegate</a>, <a href="/wiki/Robert_E._Bixby" class="mw-redirect" title="Robert E. Bixby">Robert E. Bixby</a>, Vašek Chvátal, and <a href="/wiki/William_J._Cook" title="William J. Cook">William J. Cook</a> developed one such solver, <a href="/wiki/Concorde_TSP_Solver" title="Concorde TSP Solver">Concorde</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The team was awarded The Beale-Orchard-Hays Prize for Excellence in Computational Mathematical Programming in 2000 for their ten-page paper <sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> enumerating some of Concorde's refinements of the branch and cut method that led to the solution of a 13,509-city instance and it was awarded the Frederick W. Lanchester Prize in 2007 for their book, <i>The Traveling Salesman Problem: A Computational Study</i>. </p><p>Chvátal is also known for proving the <a href="/wiki/Art_gallery_theorem" class="mw-redirect" title="Art gallery theorem">art gallery theorem</a>,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> for researching a self-describing digital sequence,<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> for his work with <a href="/wiki/David_Sankoff" title="David Sankoff">David Sankoff</a> on the <a href="/wiki/Chv%C3%A1tal%E2%80%93Sankoff_constants" title="Chvátal–Sankoff constants">Chvátal–Sankoff constants</a> controlling the behavior of the <a href="/wiki/Longest_common_subsequence_problem" class="mw-redirect" title="Longest common subsequence problem">longest common subsequence problem</a> on random inputs,<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> and for his work with <a href="/wiki/Endre_Szemer%C3%A9di" title="Endre Szemerédi">Endre Szemerédi</a> on hard instances for <a href="/wiki/Resolution_(logic)" title="Resolution (logic)">resolution theorem proving</a>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Books">Books</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=V%C3%A1clav_Chv%C3%A1tal&action=edit&section=3" title="Edit section: Books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFVašek_Chvátal1983" class="citation book cs1">Vašek Chvátal (1983). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/linearprogrammin00chv"><i>Linear Programming</i></a></span>. W.H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-1587-0" title="Special:BookSources/978-0-7167-1587-0"><bdi>978-0-7167-1587-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Programming&rft.pub=W.H.+Freeman&rft.date=1983&rft.isbn=978-0-7167-1587-0&rft.au=Va%C5%A1ek+Chv%C3%A1tal&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flinearprogrammin00chv&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>. Japanese translation published by Keigaku Shuppan, Tokyo, 1986.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC._Berge_and_V._Chvátal_(eds.)1984" class="citation book cs1"><a href="/wiki/Claude_Berge" title="Claude Berge">C. Berge</a> and V. Chvátal (eds.) (1984). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=m34PYru8s7MC"><i>Topics on Perfect Graphs</i></a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-86587-8" title="Special:BookSources/978-0-444-86587-8"><bdi>978-0-444-86587-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+on+Perfect+Graphs&rft.pub=Elsevier&rft.date=1984&rft.isbn=978-0-444-86587-8&rft.au=C.+Berge+and+V.+Chv%C3%A1tal+%28eds.%29&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dm34PYru8s7MC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|author=</code> has generic name (<a href="/wiki/Help:CS1_errors#generic_name" title="Help:CS1 errors">help</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_L._ApplegateRobert_E._BixbyVašek_ChvátalWilliam_J._Cook2007" class="citation book cs1">David L. Applegate; Robert E. Bixby; Vašek Chvátal; <a href="/wiki/William_J._Cook" title="William J. Cook">William J. Cook</a> (2007). <a rel="nofollow" class="external text" href="http://press.princeton.edu/titles/8451.html"><i>The Traveling Salesman Problem: A Computational Study</i></a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12993-8" title="Special:BookSources/978-0-691-12993-8"><bdi>978-0-691-12993-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Traveling+Salesman+Problem%3A+A+Computational+Study&rft.pub=Princeton+University+Press&rft.date=2007&rft.isbn=978-0-691-12993-8&rft.au=David+L.+Applegate&rft.au=Robert+E.+Bixby&rft.au=Va%C5%A1ek+Chv%C3%A1tal&rft.au=William+J.+Cook&rft_id=http%3A%2F%2Fpress.princeton.edu%2Ftitles%2F8451.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVašek_Chvátal2011" class="citation book cs1">Vašek Chvátal, ed. (2011). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170321082600/http://www.iospress.cn/book/combinatorial-optimization-methods-and-applications/"><i>Combinatorial Optimization: Methods and Applications</i></a>. IOS Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60750-717-8" title="Special:BookSources/978-1-60750-717-8"><bdi>978-1-60750-717-8</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://www.iospress.cn/book/combinatorial-optimization-methods-and-applications/">the original</a> on 2017-03-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-03-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorial+Optimization%3A+Methods+and+Applications&rft.pub=IOS+Press&rft.date=2011&rft.isbn=978-1-60750-717-8&rft_id=http%3A%2F%2Fwww.iospress.cn%2Fbook%2Fcombinatorial-optimization-methods-and-applications%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVašek_Chvátal2021" class="citation book cs1">Vašek Chvátal (2021). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/discrete-mathematical-charms-of-paul-erdos/7DEE9C7A7A4958CE41666F0B730E9E7B#fndtn-information"><i>Discrete Mathematical Charms of Paul Erdős. A Simple Introduction</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-92740-6" title="Special:BookSources/978-1-108-92740-6"><bdi>978-1-108-92740-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematical+Charms+of+Paul+Erd%C5%91s.+A+Simple+Introduction&rft.pub=Cambridge+University+Press&rft.date=2021&rft.isbn=978-1-108-92740-6&rft.au=Va%C5%A1ek+Chv%C3%A1tal&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fbooks%2Fdiscrete-mathematical-charms-of-paul-erdos%2F7DEE9C7A7A4958CE41666F0B730E9E7B%23fndtn-information&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=V%C3%A1clav_Chv%C3%A1tal&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_University_of_Waterloo_people" title="List of University of Waterloo people">List of University of Waterloo people</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=V%C3%A1clav_Chv%C3%A1tal&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.mathprog.org/prz/boh.htm#winners">Past Winners of The Beale-Orchard-Hays Prize</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.informs.org/Recognize-Excellence/Award-Recipients/Vasek-Chvatal">Frederick W. Lanchester Prize 2007</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160820223356/https://www.informs.org/Recognize-Excellence/Award-Recipients/Vasek-Chvatal">Archived</a> 2016-08-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, retrieved 2017-03-19.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.informs.org/Recognize-Excellence/Award-Recipients/Vasek-Chvatal">John von Neumann Theory Prize 2015</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160820223356/https://www.informs.org/Recognize-Excellence/Award-Recipients/Vasek-Chvatal">Archived</a> 2016-08-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, retrieved 2017-03-19.</span> </li> <li id="cite_note-abcr-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-abcr_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-abcr_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-abcr_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-abcr_4-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-abcr_4-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-abcr_4-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAvisBondyCookReed2007" class="citation journal cs1"><a href="/wiki/David_Avis" title="David Avis">Avis, D.</a>; Bondy, A.; <a href="/wiki/William_J._Cook" title="William J. Cook">Cook, W.</a>; Reed, B. (2007). <a rel="nofollow" class="external text" href="http://www.math.uwaterloo.ca/~bico//papers/vasek_gc.pdf">"Vasek Chvatal: A Short Introduction"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Graphs_and_Combinatorics" title="Graphs and Combinatorics">Graphs and Combinatorics</a></i>. <b>23</b>: 41–66. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.127.5910">10.1.1.127.5910</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00373-007-0721-4">10.1007/s00373-007-0721-4</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11121944">11121944</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Graphs+and+Combinatorics&rft.atitle=Vasek+Chvatal%3A+A+Short+Introduction&rft.volume=23&rft.pages=41-66&rft.date=2007&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.127.5910%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11121944%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00373-007-0721-4&rft.aulast=Avis&rft.aufirst=D.&rft.au=Bondy%2C+A.&rft.au=Cook%2C+W.&rft.au=Reed%2C+B.&rft_id=http%3A%2F%2Fwww.math.uwaterloo.ca%2F~bico%2F%2Fpapers%2Fvasek_gc.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span></span> </li> <li id="cite_note-travprof-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-travprof_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-travprof_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://ctr.concordia.ca/2004-05/feb_10/01/index.shtml">Vasek Chvátal is ‘the travelling professor’</a>, Concordia's Thursday Report, Feb. 10, 2005.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=37109">The Mathematics Genealogy Project – Václav Chvátal</a></span> </li> <li id="cite_note-canchair-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-canchair_7-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://ctr.concordia.ca/2003-04/oct_23/01-chvatal/index.shtml">Vasek Chvatal awarded Canada Research Chair</a>, Concordia's Thursday Report, Oct. 23, 2003.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátal1997" class="citation cs2 cs1-prop-long-vol"><a class="mw-selflink selflink">Chvátal, Vašek</a> (1997), "In praise of Claude Berge", <i><a href="/wiki/Discrete_Mathematics_(journal)" title="Discrete Mathematics (journal)">Discrete Mathematics</a></i>, 165–166: 3–9, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0012-365x%2896%2900156-2">10.1016/s0012-365x(96)00156-2</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=In+praise+of+Claude+Berge&rft.volume=165-166&rft.pages=3-9&rft.date=1997&rft_id=info%3Adoi%2F10.1016%2Fs0012-365x%2896%2900156-2&rft.aulast=Chv%C3%A1tal&rft.aufirst=Va%C5%A1ek&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>,</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátal1965" class="citation cs2"><a class="mw-selflink selflink">Chvátal, Václav</a> (1965), <a rel="nofollow" class="external text" href="http://dml.cz/dmlcz/105034">"On finite and countable rigid graphs and tournaments"</a>, <i><a href="/w/index.php?title=Commentationes_Mathematicae_Universitatis_Carolinae&action=edit&redlink=1" class="new" title="Commentationes Mathematicae Universitatis Carolinae (page does not exist)">Commentationes Mathematicae Universitatis Carolinae</a></i>, <b>6</b>: 429–438</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Commentationes+Mathematicae+Universitatis+Carolinae&rft.atitle=On+finite+and+countable+rigid+graphs+and+tournaments&rft.volume=6&rft.pages=429-438&rft.date=1965&rft.aulast=Chv%C3%A1tal&rft.aufirst=V%C3%A1clav&rft_id=http%3A%2F%2Fdml.cz%2Fdmlcz%2F105034&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Chvátal_Graph"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ChvatalGraph.html">"Chvátal Graph"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Chv%C3%A1tal+Graph&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FChvatalGraph.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFV._ChvátalP._Erdős1972" class="citation cs2"><a class="mw-selflink selflink">V. Chvátal</a>; <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">P. Erdős</a> (1972), <a rel="nofollow" class="external text" href="http://www.math-inst.hu/~p_erdos/1972-02.pdf">"A note on Hamiltonian circuits"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Discrete_Mathematics_(journal)" title="Discrete Mathematics (journal)">Discrete Mathematics</a></i>, <b>2</b> (2): 111–113, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365x%2872%2990079-9">10.1016/0012-365x(72)90079-9</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=A+note+on+Hamiltonian+circuits&rft.volume=2&rft.issue=2&rft.pages=111-113&rft.date=1972&rft_id=info%3Adoi%2F10.1016%2F0012-365x%2872%2990079-9&rft.au=V.+Chv%C3%A1tal&rft.au=P.+Erd%C5%91s&rft_id=http%3A%2F%2Fwww.math-inst.hu%2F~p_erdos%2F1972-02.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>,</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátal1973" class="citation cs2"><a class="mw-selflink selflink">Chvátal, V.</a> (1973), "Tough graphs and hamiltonian circuits", <i><a href="/wiki/Discrete_Mathematics_(journal)" title="Discrete Mathematics (journal)">Discrete Mathematics</a></i>, <b>5</b> (3): 215–228, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365x%2873%2990138-6">10.1016/0012-365x(73)90138-6</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=Tough+graphs+and+hamiltonian+circuits&rft.volume=5&rft.issue=3&rft.pages=215-228&rft.date=1973&rft_id=info%3Adoi%2F10.1016%2F0012-365x%2873%2990138-6&rft.aulast=Chv%C3%A1tal&rft.aufirst=V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>,</span> </li> <li id="cite_note-lesniak-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-lesniak_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLesniak" class="citation cs2">Lesniak, Linda, <a rel="nofollow" class="external text" href="http://faculty.nps.edu/rgera/Conjectures/jmm2012/Linda-Lesniak.pdf"><i>Chvátal's t<sub>0</sub>-tough conjecture</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chv%C3%A1tal%27s+t%3Csub%3E0%3C%2Fsub%3E-tough+conjecture&rft.aulast=Lesniak&rft.aufirst=Linda&rft_id=http%3A%2F%2Ffaculty.nps.edu%2Frgera%2FConjectures%2Fjmm2012%2FLinda-Lesniak.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Mathematical Reviews MR0369170</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFV._ChvátalDavid_A._KlarnerD.E._Knuth1972" class="citation cs2 cs1-prop-long-vol"><a class="mw-selflink selflink">V. 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Knuth</a> (1972), <a rel="nofollow" class="external text" href="http://users.encs.concordia.ca/~chvatal/Stan-CS-TR-72-292.pdf">"Selected combinatorial research problems"</a> <span class="cs1-format">(PDF)</span>, <i>Computer Science Department, Stanford University</i>, Stan-CS-TR-72-292</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Science+Department%2C+Stanford+University&rft.atitle=Selected+combinatorial+research+problems&rft.volume=Stan-CS-TR-72-292&rft.date=1972&rft.au=V.+Chv%C3%A1tal&rft.au=David+A.+Klarner&rft.au=D.E.+Knuth&rft_id=http%3A%2F%2Fusers.encs.concordia.ca%2F~chvatal%2FStan-CS-TR-72-292.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>: Problem 25</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátal" class="citation cs2"><a class="mw-selflink selflink">Chvátal, Vašek</a>, <a rel="nofollow" class="external text" href="http://users.encs.concordia.ca/~chvatal/conjecture.html"><i>A conjecture in extremal combinatorics</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+conjecture+in+extremal+combinatorics&rft.aulast=Chv%C3%A1tal&rft.aufirst=Va%C5%A1ek&rft_id=http%3A%2F%2Fusers.encs.concordia.ca%2F~chvatal%2Fconjecture.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">"A greedy heuristic for the set-covering problem", Mathematics of Operations Research, 1979</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátal1973" class="citation cs2"><a class="mw-selflink selflink">Chvátal, Václav</a> (1973), "Edmonds polytopes and weakly hamiltonian graphs", <i>Mathematical Programming</i>, <b>5</b>: 29–40, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01580109">10.1007/BF01580109</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8140217">8140217</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Programming&rft.atitle=Edmonds+polytopes+and+weakly+hamiltonian+graphs&rft.volume=5&rft.pages=29-40&rft.date=1973&rft_id=info%3Adoi%2F10.1007%2FBF01580109&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8140217%23id-name%3DS2CID&rft.aulast=Chv%C3%A1tal&rft.aufirst=V%C3%A1clav&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>,</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátal1973" class="citation cs2"><a class="mw-selflink selflink">Chvátal, Václav</a> (1973), "Edmonds polytopes and a hierarchy of combinatorial problems", <i>Discrete Mathematics</i>, <b>4</b> (4): 305–337, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365x%2873%2990167-2">10.1016/0012-365x(73)90167-2</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=Edmonds+polytopes+and+a+hierarchy+of+combinatorial+problems&rft.volume=4&rft.issue=4&rft.pages=305-337&rft.date=1973&rft_id=info%3Adoi%2F10.1016%2F0012-365x%2873%2990167-2&rft.aulast=Chv%C3%A1tal&rft.aufirst=V%C3%A1clav&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>,</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátal1975" class="citation cs2"><a class="mw-selflink selflink">Chvátal, Václav</a> (1975), <a rel="nofollow" class="external text" href="http://i.stanford.edu/pub/cstr/reports/cs/tr/75/505/CS-TR-75-505.pdf">"Some linear programming aspects of combinatorics"</a> <span class="cs1-format">(PDF)</span>, <i>Congressus Numerantium</i>, <b>13</b>: 2–30</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Congressus+Numerantium&rft.atitle=Some+linear+programming+aspects+of+combinatorics&rft.volume=13&rft.pages=2-30&rft.date=1975&rft.aulast=Chv%C3%A1tal&rft.aufirst=V%C3%A1clav&rft_id=http%3A%2F%2Fi.stanford.edu%2Fpub%2Fcstr%2Freports%2Fcs%2Ftr%2F75%2F505%2FCS-TR-75-505.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>,</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátal1975" class="citation cs2"><a class="mw-selflink selflink">Chvátal, V.</a> (1975), "On certain polytopes associated with graphs", <i>Journal of Combinatorial Theory, Series B</i>, <b>18</b> (2): 138–154, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0095-8956%2875%2990041-6">10.1016/0095-8956(75)90041-6</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Combinatorial+Theory%2C+Series+B&rft.atitle=On+certain+polytopes+associated+with+graphs&rft.volume=18&rft.issue=2&rft.pages=138-154&rft.date=1975&rft_id=info%3Adoi%2F10.1016%2F0095-8956%2875%2990041-6&rft.aulast=Chv%C3%A1tal&rft.aufirst=V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Math Problem, Long Baffling, Slowly Yields. <a href="/wiki/New_York_Times" class="mw-redirect" title="New York Times">New York Times</a>, Mar. 12, 1991.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.sciencenews.org/articles/20050101/mathtrek.asp">Artful Routes</a>, Science News Online, Jan. 1, 2005.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApplegateBixbyChvátalCook1998" class="citation cs2 cs1-prop-long-vol">Applegate, David; Bixby, Robert; <a class="mw-selflink selflink">Chvátal, Vašek</a>; <a href="/wiki/William_J._Cook" title="William J. 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Art gallery problems</a>, AMS Feature Column by Joseph Malkevitch</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Combinatorics/Chvatal.shtml">Chvatal's Art Gallery Theorem</a> in <a href="/wiki/Alexander_Bogomolny" title="Alexander Bogomolny">Alexander Bogomolny</a>'s Cut the Knot</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.tv.com/shows/numb3rs/obsession-495186/trivia/">Obsession</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170320145044/http://www.tv.com/shows/numb3rs/obsession-495186/trivia/">Archived</a> 2017-03-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Numb3rs, Episode 3, Season 2</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátal1993" class="citation cs2 cs1-prop-long-vol"><a class="mw-selflink selflink">Chvátal, Vašek</a> (1993), <a rel="nofollow" class="external text" href="http://dimacs.rutgers.edu/TechnicalReports/abstracts/1993/93-84.html">"Notes on the Kolakoski Sequence"</a>, <i>DIMACS Technical Reports</i>, TR: 93-84</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=DIMACS+Technical+Reports&rft.atitle=Notes+on+the+Kolakoski+Sequence&rft.volume=TR%3A+93-84&rft.date=1993&rft.aulast=Chv%C3%A1tal&rft.aufirst=Va%C5%A1ek&rft_id=http%3A%2F%2Fdimacs.rutgers.edu%2FTechnicalReports%2Fabstracts%2F1993%2F93-84.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.sciencenews.org/article/dangerous-problems-0">Dangerous Problems</a>, Science News Online, Jul. 13, 2002.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátalSankoff1975" class="citation cs2"><a class="mw-selflink selflink">Chvátal, Václav</a>; <a href="/wiki/David_Sankoff" title="David Sankoff">Sankoff, David</a> (1975), "Longest common subsequences of two random sequences", <i>Journal of Applied Probability</i>, <b>12</b> (2): 306–315, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3212444">10.2307/3212444</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3212444">3212444</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250345191">250345191</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Probability&rft.atitle=Longest+common+subsequences+of+two+random+sequences&rft.volume=12&rft.issue=2&rft.pages=306-315&rft.date=1975&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250345191%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3212444%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3212444&rft.aulast=Chv%C3%A1tal&rft.aufirst=V%C3%A1clav&rft.au=Sankoff%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChvátalSzemerédi1988" class="citation cs2"><a class="mw-selflink selflink">Chvátal, Vašek</a>; <a href="/wiki/Endre_Szemer%C3%A9di" title="Endre Szemerédi">Szemerédi, Endre</a> (1988), "Many hard examples for resolution", <i>Journal of the ACM</i>, <b>35</b> (4): 759–768, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F48014.48016">10.1145/48014.48016</a></span>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2526816">2526816</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+ACM&rft.atitle=Many+hard+examples+for+resolution&rft.volume=35&rft.issue=4&rft.pages=759-768&rft.date=1988&rft_id=info%3Adoi%2F10.1145%2F48014.48016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2526816%23id-name%3DS2CID&rft.aulast=Chv%C3%A1tal&rft.aufirst=Va%C5%A1ek&rft.au=Szemer%C3%A9di%2C+Endre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span>.</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorchers,_Brian2007" class="citation web cs1">Borchers, Brian (March 25, 2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230423174407/https://www.maa.org/press/maa-reviews/the-traveling-salesman-problem-a-computational-study">"Review of <i>The Traveling Salesman Problem: A Computational Study</i>"</a>. <i>MAA Reviews, Mathematical Association of America</i>. Archived from <a rel="nofollow" class="external text" href="https://www.maa.org/press/maa-reviews/the-traveling-salesman-problem-a-computational-study">the original</a> on April 23, 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">June 21,</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MAA+Reviews%2C+Mathematical+Association+of+America&rft.atitle=Review+of+The+Traveling+Salesman+Problem%3A+A+Computational+Study&rft.date=2007-03-25&rft.au=Borchers%2C+Brian&rft_id=https%3A%2F%2Fwww.maa.org%2Fpress%2Fmaa-reviews%2Fthe-traveling-salesman-problem-a-computational-study&rfr_id=info%3Asid%2Fen.wikipedia.org%3AV%C3%A1clav+Chv%C3%A1tal" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=V%C3%A1clav_Chv%C3%A1tal&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" 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style="font-size:114%;margin:0 4em"><a href="/wiki/John_von_Neumann_Theory_Prize" title="John von Neumann Theory Prize">John von Neumann Theory Prize</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">1975–1999</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/George_Dantzig" title="George Dantzig">George Dantzig</a> <small>(1975)</small></li> <li><a href="/wiki/Richard_Bellman" class="mw-redirect" title="Richard Bellman">Richard Bellman</a> <small>(1976)</small></li> <li><a href="/wiki/Felix_Pollaczek" class="mw-redirect" title="Felix Pollaczek">Felix Pollaczek</a> <small>(1977)</small></li> <li><a href="/wiki/John_Forbes_Nash_Jr." title="John Forbes Nash Jr.">John F. Nash</a> / <a href="/wiki/Carlton_E._Lemke" title="Carlton E. Lemke">Carlton E. Lemke</a> <small>(1978)</small></li> <li><a href="/wiki/David_Blackwell" title="David Blackwell">David Blackwell</a> <small>(1979)</small></li> <li><a href="/wiki/David_Gale" title="David Gale">David Gale</a> / <a href="/wiki/Harold_W._Kuhn" title="Harold W. Kuhn">Harold W. Kuhn</a> / <a href="/wiki/Albert_W._Tucker" title="Albert W. Tucker">Albert W. Tucker</a> <small>(1980)</small></li> <li><a href="/wiki/Lloyd_Shapley" title="Lloyd Shapley">Lloyd Shapley</a> <small>(1981)</small></li> <li><a href="/wiki/Abraham_Charnes" title="Abraham Charnes">Abraham Charnes</a> / <a href="/wiki/William_W._Cooper" title="William W. Cooper">William W. Cooper</a> / <a href="/wiki/Richard_J._Duffin" class="mw-redirect" title="Richard J. Duffin">Richard J. Duffin</a> <small>(1982)</small></li> <li><a href="/wiki/Herbert_Scarf" title="Herbert Scarf">Herbert Scarf</a> <small>(1983)</small></li> <li><a href="/wiki/Ralph_Gomory" class="mw-redirect" title="Ralph Gomory">Ralph Gomory</a> <small>(1984)</small></li> <li><a href="/wiki/Jack_Edmonds" title="Jack Edmonds">Jack Edmonds</a> <small>(1985)</small></li> <li><a href="/wiki/Kenneth_Arrow" title="Kenneth Arrow">Kenneth Arrow</a> <small>(1986)</small></li> <li><a href="/wiki/Samuel_Karlin" title="Samuel Karlin">Samuel Karlin</a> <small>(1987)</small></li> <li><a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Herbert A. Simon</a> <small>(1988)</small></li> <li><a href="/wiki/Harry_Markowitz" title="Harry Markowitz">Harry Markowitz</a> <small>(1989)</small></li> <li><a href="/wiki/Richard_Karp" class="mw-redirect" title="Richard Karp">Richard Karp</a> <small>(1990)</small></li> <li><a href="/wiki/Richard_E._Barlow" title="Richard E. Barlow">Richard E. Barlow</a> / <a href="/w/index.php?title=Frank_Proschan&action=edit&redlink=1" class="new" title="Frank Proschan (page does not exist)">Frank Proschan</a> <small>(1991)</small></li> <li><a href="/wiki/Alan_J._Hoffman" title="Alan J. Hoffman">Alan J. Hoffman</a> / <a href="/wiki/Philip_Wolfe_(mathematician)" title="Philip Wolfe (mathematician)">Philip Wolfe</a> <small>(1992)</small></li> <li><a href="/wiki/Robert_Herman" title="Robert Herman">Robert Herman</a> <small>(1993)</small></li> <li><a href="/wiki/Lajos_Takacs" class="mw-redirect" title="Lajos Takacs">Lajos Takacs</a> <small>(1994)</small></li> <li><a href="/wiki/Egon_Balas" title="Egon Balas">Egon Balas</a> <small>(1995)</small></li> <li><a href="/wiki/Peter_C._Fishburn" title="Peter C. Fishburn">Peter C. Fishburn</a> <small>(1996)</small></li> <li><a href="/wiki/Peter_Whittle_(mathematician)" title="Peter Whittle (mathematician)">Peter Whittle</a> <small>(1997)</small></li> <li><a href="/wiki/Fred_W._Glover" title="Fred W. Glover">Fred W. Glover</a> <small>(1998)</small></li> <li><a href="/wiki/R._Tyrrell_Rockafellar" title="R. Tyrrell Rockafellar">R. Tyrrell Rockafellar</a> <small>(1999)</small></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">2000–present</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ellis_L._Johnson" title="Ellis L. Johnson">Ellis L. Johnson</a> / <a href="/wiki/Manfred_W._Padberg" title="Manfred W. Padberg">Manfred W. Padberg</a> <small>(2000)</small></li> <li><a href="/wiki/Ward_Whitt" title="Ward Whitt">Ward Whitt</a> <small>(2001)</small></li> <li><a href="/w/index.php?title=Donald_L._Iglehart&action=edit&redlink=1" class="new" title="Donald L. Iglehart (page does not exist)">Donald L. Iglehart</a> / <a href="/wiki/Cyrus_Derman" title="Cyrus Derman">Cyrus Derman</a> <small>(2002)</small></li> <li><a href="/wiki/Arkadi_Nemirovski" title="Arkadi Nemirovski">Arkadi Nemirovski</a> / <a href="/w/index.php?title=Michael_Jeremy_Todd&action=edit&redlink=1" class="new" title="Michael Jeremy Todd (page does not exist)">Michael J. Todd</a> <small>(2003)</small></li> <li><a href="/wiki/J._Michael_Harrison" title="J. Michael Harrison">J. Michael Harrison</a> <small>(2004)</small></li> <li><a href="/wiki/Robert_Aumann" title="Robert Aumann">Robert Aumann</a> <small>(2005)</small></li> <li><a href="/wiki/Martin_Gr%C3%B6tschel" title="Martin Grötschel">Martin Grötschel</a> / <a href="/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">László Lovász</a> / <a href="/wiki/Alexander_Schrijver" title="Alexander Schrijver">Alexander Schrijver</a> <small>(2006)</small></li> <li><a href="/w/index.php?title=Arthur_F._Veinott,_Jr.&action=edit&redlink=1" class="new" title="Arthur F. Veinott, Jr. (page does not exist)">Arthur F. Veinott, Jr.</a> <small>(2007)</small></li> <li><a href="/wiki/Frank_Kelly_(mathematician)" title="Frank Kelly (mathematician)">Frank Kelly</a> <small>(2008)</small></li> <li><a href="/wiki/Yurii_Nesterov" title="Yurii Nesterov">Yurii Nesterov</a> / <a href="/wiki/Yinyu_Ye" title="Yinyu Ye">Yinyu Ye</a> <small>(2009)</small></li> <li><a href="/w/index.php?title=S%C3%B8ren_Asmussen&action=edit&redlink=1" class="new" title="Søren Asmussen (page does not exist)">Søren Asmussen</a> / <a href="/w/index.php?title=Peter_W._Glynn&action=edit&redlink=1" class="new" title="Peter W. Glynn (page does not exist)">Peter W. Glynn</a> <small>(2010)</small></li> <li><a href="/wiki/G%C3%A9rard_Cornu%C3%A9jols" title="Gérard Cornuéjols">Gérard Cornuéjols</a> <small>(2011)</small></li> <li><a href="/wiki/George_Nemhauser" title="George Nemhauser">George Nemhauser</a> / <a href="/wiki/Laurence_Wolsey" title="Laurence Wolsey">Laurence Wolsey</a> <small>(2012)</small></li> <li><a href="/wiki/Michel_Balinski" title="Michel Balinski">Michel Balinski</a> <small>(2013)</small></li> <li><a href="/wiki/Nimrod_Megiddo" title="Nimrod Megiddo">Nimrod Megiddo</a> <small>(2014)</small></li> <li><a href="/wiki/Va%C5%A1ek_Chv%C3%A1tal" class="mw-redirect" title="Vašek Chvátal">Vašek Chvátal</a> / <a href="/w/index.php?title=Jean_Bernard_Lasserre&action=edit&redlink=1" class="new" title="Jean Bernard Lasserre (page does not exist)">Jean Bernard Lasserre</a> <small>(2015)</small></li> <li><a href="/wiki/Martin_I._Reiman" title="Martin I. Reiman">Martin I. Reiman</a> / <a href="/wiki/Ruth_J._Williams" title="Ruth J. Williams">Ruth J. Williams</a> <small>(2016)</small></li> <li><a href="/wiki/Donald_Goldfarb" title="Donald Goldfarb">Donald Goldfarb</a> / <a href="/wiki/Jorge_Nocedal" title="Jorge Nocedal">Jorge Nocedal</a> <small>(2017)</small></li> <li><a href="/wiki/Dimitri_Bertsekas" title="Dimitri Bertsekas">Dimitri Bertsekas</a> / <a href="/wiki/John_Tsitsiklis" title="John Tsitsiklis">John Tsitsiklis</a> <small>(2018)</small></li> <li><a href="/wiki/Dimitris_Bertsimas" title="Dimitris Bertsimas">Dimitris Bertsimas</a> / <a href="/w/index.php?title=Jong-Shi_Pang&action=edit&redlink=1" class="new" title="Jong-Shi Pang (page does not exist)">Jong-Shi Pang</a> <small>(2019)</small></li> <li><a href="/wiki/Adrian_Lewis_(mathematician)" title="Adrian Lewis (mathematician)">Adrian Lewis</a> <small>(2020)</small></li> <li><a href="/wiki/Alexander_Shapiro" title="Alexander Shapiro">Alexander Shapiro</a> <small>(2021)</small></li> <li><a href="/wiki/Vijay_Vazirani" title="Vijay Vazirani">Vijay Vazirani</a> <small>(2022)</small></li> <li><a href="/wiki/Christos_Papadimitriou" title="Christos Papadimitriou">Christos Papadimitriou</a> / <a href="/wiki/Mihalis_Yannakakis" title="Mihalis Yannakakis">Mihalis Yannakakis</a> <small>(2023)</small></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1384912#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1384912#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q1384912#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, 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style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/110721772">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/n82157087">United States</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb123592372">France</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb123592372">BnF data</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=jcu2017956904&CON_LNG=ENG">Czech Republic</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://data.bibliotheken.nl/id/thes/p071295607">Netherlands</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007458178105171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Academics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.mathgenealogy.org/id.php?id=37109">Mathematics Genealogy Project</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://dl.acm.org/profile/81100012354">Association for Computing Machinery</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://zbmath.org/authors/?q=ai:chvatal.vaclav">zbMATH</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://dblp.org/pid/c/VasekChvatal">DBLP</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet/MRAuthorID/49270">MathSciNet</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.idref.fr/032591950">IdRef</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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