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Available in 37 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-37" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">37 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%B3%D9%85_%D8%A8%D9%8A%D8%A7%D9%86%D9%8A_%D9%83%D8%A7%D9%85%D9%84" title="رسم بياني كامل – Arabic" lang="ar" hreflang="ar" data-title="رسم بياني كامل" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Graf_complet" title="Graf complet – Catalan" lang="ca" hreflang="ca" data-title="Graf complet" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C3%9Apln%C3%BD_graf" title="Úplný graf – Czech" lang="cs" hreflang="cs" data-title="Úplný graf" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Komplet_graf" title="Komplet graf – Danish" lang="da" hreflang="da" data-title="Komplet graf" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Vollst%C3%A4ndiger_Graph" title="Vollständiger Graph – German" lang="de" hreflang="de" data-title="Vollständiger Graph" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/T%C3%A4isgraaf" title="Täisgraaf – Estonian" lang="et" hreflang="et" data-title="Täisgraaf" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BB%CE%AE%CF%81%CE%B7%CF%82_%CE%B3%CF%81%CE%AC%CF%86%CE%BF%CF%82" title="Πλήρης γράφος – Greek" lang="el" hreflang="el" data-title="Πλήρης γράφος" 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/Grafo_completo" title="Grafo completo – Spanish" lang="es" hreflang="es" data-title="Grafo completo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Plena_grafeo" title="Plena grafeo – Esperanto" lang="eo" hreflang="eo" data-title="Plena grafeo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Grafo_oso" title="Grafo oso – Basque" lang="eu" hreflang="eu" data-title="Grafo oso" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D8%A7%D9%81_%DA%A9%D8%A7%D9%85%D9%84" title="گراف کامل – Persian" lang="fa" hreflang="fa" data-title="گراف کامل" 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/Graphe_complet" title="Graphe complet – French" lang="fr" hreflang="fr" data-title="Graphe complet" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%99%84%EC%A0%84_%EA%B7%B8%EB%9E%98%ED%94%84" title="완전 그래프 – Korean" lang="ko" hreflang="ko" data-title="완전 그래프" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Potpun_graf" title="Potpun graf – Croatian" lang="hr" hreflang="hr" data-title="Potpun graf" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Graf_lengkap" title="Graf lengkap – Indonesian" lang="id" hreflang="id" data-title="Graf lengkap" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fulltengt_net" title="Fulltengt net – Icelandic" lang="is" hreflang="is" data-title="Fulltengt net" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Grafo_completo" title="Grafo completo – Italian" lang="it" hreflang="it" data-title="Grafo completo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%A8%D7%A3_%D7%A9%D7%9C%D7%9D" title="גרף שלם – Hebrew" lang="he" hreflang="he" data-title="גרף שלם" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BB%D1%8B%D2%9B_%D0%B3%D1%80%D0%B0%D1%84" title="Толық граф – Kazakh" lang="kk" hreflang="kk" data-title="Толық граф" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pilnasis_grafas" title="Pilnasis grafas – Lithuanian" lang="lt" hreflang="lt" data-title="Pilnasis grafas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Teljes_gr%C3%A1f" title="Teljes gráf – Hungarian" lang="hu" hreflang="hu" data-title="Teljes gráf" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E3%82%B0%E3%83%A9%E3%83%95" title="完全グラフ – Japanese" lang="ja" hreflang="ja" data-title="完全グラフ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Graf_pe%C5%82ny" title="Graf pełny – Polish" lang="pl" hreflang="pl" data-title="Graf pełny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grafo_completo" title="Grafo completo – Portuguese" lang="pt" hreflang="pt" data-title="Grafo completo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Graf_complet" title="Graf complet – Romanian" lang="ro" hreflang="ro" data-title="Graf complet" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%BD%D1%8B%D0%B9_%D0%B3%D1%80%D0%B0%D1%84" title="Полный граф – Russian" lang="ru" hreflang="ru" data-title="Полный граф" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/%C3%9Apln%C3%BD_graf" title="Úplný graf – Slovak" lang="sk" hreflang="sk" data-title="Úplný graf" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Polni_graf" title="Polni graf – Slovenian" lang="sl" hreflang="sl" data-title="Polni graf" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D1%82%D0%B0%D0%BD_%D0%B3%D1%80%D0%B0%D1%84" title="Комплетан граф – Serbian" lang="sr" hreflang="sr" data-title="Комплетан граф" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Komplett_graf" title="Komplett graf – Swedish" lang="sv" hreflang="sv" data-title="Komplett graf" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%B4%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%81%E0%AE%B0%E0%AF%81" title="முழுக்கோட்டுரு – Tamil" lang="ta" hreflang="ta" data-title="முழுக்கோட்டுரு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A3%E0%B8%B2%E0%B8%9F%E0%B8%9A%E0%B8%A3%E0%B8%B4%E0%B8%9A%E0%B8%B9%E0%B8%A3%E0%B8%93%E0%B9%8C" title="กราฟบริบูรณ์ – Thai" lang="th" hreflang="th" data-title="กราฟบริบูรณ์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D0%BD%D0%B8%D0%B9_%D0%B3%D1%80%D0%B0%D1%84" title="Повний граф – Ukrainian" lang="uk" hreflang="uk" data-title="Повний граф" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%DA%A9%D9%85%D9%84_%DA%AF%D8%B1%D8%A7%D9%81" title="مکمل گراف – Urdu" lang="ur" hreflang="ur" data-title="مکمل گراف" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%93_th%E1%BB%8B_%C4%91%E1%BA%A7y_%C4%91%E1%BB%A7" title="Đồ thị đầy đủ – Vietnamese" lang="vi" hreflang="vi" data-title="Đồ thị đầy đủ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E5%9C%96" title="完全圖 – Cantonese" lang="yue" hreflang="yue" data-title="完全圖" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E5%9C%96" title="完全圖 – Chinese" lang="zh" hreflang="zh" data-title="完全圖" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q45715#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav 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.infobox-image img{background-color:var(--background-color-inverted,#f8f9fa)}}</style><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"><tbody><tr><th colspan="2" class="infobox-above">Complete graph</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Complete_graph_K7.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Complete_graph_K7.svg/200px-Complete_graph_K7.svg.png" decoding="async" width="200" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Complete_graph_K7.svg/300px-Complete_graph_K7.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Complete_graph_K7.svg/400px-Complete_graph_K7.svg.png 2x" data-file-width="10552" data-file-height="10352" /></a></span><div class="infobox-caption"><span class="texhtml"><i>K</i><sub>7</sub></span>, a complete graph with 7 vertices</div></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;"><a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">Vertices</a></th><td class="infobox-data"><span class="texhtml mvar" style="font-style:italic;">n</span></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;"><a href="/wiki/Edge_(graph_theory)" class="mw-redirect" title="Edge (graph theory)">Edges</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {n(n-1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {n(n-1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d038eba5d8d1c39619fbbeddaa58d23e6781e7e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.188ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {n(n-1)}{2}}}" /></span></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;"><a href="/wiki/Distance_(graph_theory)" title="Distance (graph theory)">Radius</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>n</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c5251bcc7a82d08331ca2c47c0917a9bab5d52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.557ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}" /></span></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;"><a href="/wiki/Diameter_(graph_theory)" title="Diameter (graph theory)">Diameter</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>n</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c5251bcc7a82d08331ca2c47c0917a9bab5d52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.557ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}" /></span></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;"><a href="/wiki/Girth_(graph_theory)" title="Girth (graph theory)">Girth</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{ll}\infty &n\leq 2\\3&{\text{otherwise}}\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi mathvariant="normal">∞</mi> </mtd> <mtd> <mi>n</mi> <mo>≤</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{ll}\infty &n\leq 2\\3&{\text{otherwise}}\end{array}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30dfc7545c7276b581e69d69a2b83651bdcacb7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.718ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{array}{ll}\infty &n\leq 2\\3&{\text{otherwise}}\end{array}}\right.}" /></span></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;"><a href="/wiki/Graph_automorphism" title="Graph automorphism">Automorphisms</a></th><td class="infobox-data"><span class="texhtml"><i>n</i>! (<a href="/wiki/Symmetric_group" title="Symmetric group"><i>S</i></a><sub><i>n</i></sub>)</span></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;"><a href="/wiki/Chromatic_number" class="mw-redirect" title="Chromatic number">Chromatic number</a></th><td class="infobox-data"><span class="texhtml mvar" style="font-style:italic;">n</span></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;"><a href="/wiki/Chromatic_index" class="mw-redirect" title="Chromatic index">Chromatic index</a></th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"><ul><li><span class="texhtml mvar" style="font-style:italic;">n</span> if <span class="texhtml mvar" style="font-style:italic;">n</span> is odd</li><li><span class="texhtml"><i>n</i> − 1</span> if <span class="texhtml mvar" style="font-style:italic;">n</span> is even</li></ul></div></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;"><a href="/wiki/Spectral_graph_theory" title="Spectral graph theory">Spectrum</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{lll}\emptyset &n=0\\\left\{0^{1}\right\}&n=1\\\left\{(n-1)^{1},-1^{n-1}\right\}&{\text{otherwise}}\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi mathvariant="normal">∅</mi> </mtd> <mtd> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>{</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>}</mo> </mrow> </mtd> <mtd> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>−</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>}</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{lll}\emptyset &n=0\\\left\{0^{1}\right\}&n=1\\\left\{(n-1)^{1},-1^{n-1}\right\}&{\text{otherwise}}\end{array}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26770ada60c4e2bf120f25ed26bb0fd892c90d46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:33.012ex; height:10.343ex;" alt="{\displaystyle \left\{{\begin{array}{lll}\emptyset &n=0\\\left\{0^{1}\right\}&n=1\\\left\{(n-1)^{1},-1^{n-1}\right\}&{\text{otherwise}}\end{array}}\right.}" /></span></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;">Properties</th><td class="infobox-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist"><ul><li><a href="/wiki/Regular_graph" title="Regular graph"><span class="texhtml">(<i>n</i> − 1)</span>-regular</a></li><li><a href="/wiki/Symmetric_graph" title="Symmetric graph">Symmetric graph</a></li><li><a href="/wiki/Vertex-transitive_graph" title="Vertex-transitive graph">Vertex-transitive</a></li><li><a href="/wiki/Edge-transitive_graph" title="Edge-transitive graph">Edge-transitive</a></li><li><a href="/wiki/Strongly_regular_graph" title="Strongly regular graph">Strongly regular</a></li><li><a href="/wiki/Integral_graph" title="Integral graph">Integral</a></li></ul></div></td></tr><tr><th scope="row" class="infobox-label" style="width:50%;">Notation</th><td class="infobox-data"><span class="texhtml"><i>K<sub>n</sub></i></span></td></tr><tr><td colspan="2" class="infobox-below"><a href="/wiki/List_of_graphs_by_edges_and_vertices" title="List of graphs by edges and vertices">Table of graphs and parameters</a></td></tr></tbody></table> <p>In the <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> field of <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, a <b>complete graph</b> is a <a href="/wiki/Simple_graph" class="mw-redirect" title="Simple graph">simple</a> <a href="/wiki/Undirected_graph" class="mw-redirect" title="Undirected graph">undirected graph</a> in which every pair of distinct <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertices</a> is connected by a unique <a href="/wiki/Edge_(graph_theory)" class="mw-redirect" title="Edge (graph theory)">edge</a>. A <b>complete digraph</b> is a <a href="/wiki/Directed_graph" title="Directed graph">directed graph</a> in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).<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> </p><p>Graph theory itself is typically dated as beginning with <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>'s 1736 work on the <a href="/wiki/Seven_Bridges_of_K%C3%B6nigsberg" title="Seven Bridges of Königsberg">Seven Bridges of Königsberg</a>. However, <a href="/wiki/Graph_drawing" title="Graph drawing">drawings</a> of complete graphs, with their vertices placed on the points of a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a>, had already appeared in the 13th century, in the work of <a href="/wiki/Ramon_Llull" title="Ramon Llull">Ramon Llull</a>.<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> Such a drawing is sometimes referred to as a <b>mystic rose</b>.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complete_graph&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The complete graph on <span class="texhtml mvar" style="font-style:italic;">n</span> vertices is denoted by <span class="texhtml mvar" style="font-style:italic;">K<sub>n</sub></span>. Some sources claim that the letter <span class="texhtml mvar" style="font-style:italic;">K</span> in this notation stands for the German word <span title="German-language text"><i lang="de">komplett</i></span>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> but the German name for a complete graph, <span title="German-language text"><i lang="de">vollständiger Graph</i></span>, does not contain the letter <span class="texhtml mvar" style="font-style:italic;">K</span>, and other sources state that the notation honors the contributions of <a href="/wiki/Kazimierz_Kuratowski" title="Kazimierz Kuratowski">Kazimierz Kuratowski</a> to graph theory.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><span class="texhtml mvar" style="font-style:italic;">K<sub>n</sub></span> has <span class="texhtml"><i>n</i>(<i>n</i> – 1)/2</span> edges (a <a href="/wiki/Triangular_number" title="Triangular number">triangular number</a>), and is a <a href="/wiki/Regular_graph" title="Regular graph">regular graph</a> of <a href="/wiki/Degree_(graph_theory)" title="Degree (graph theory)">degree</a> <span class="texhtml"><i>n</i> – 1</span>. All complete graphs are their own <a href="/wiki/Clique_(graph_theory)" title="Clique (graph theory)">maximal cliques</a>. They are maximally <a href="/wiki/Connectivity_(graph_theory)" title="Connectivity (graph theory)">connected</a> as the only <a href="/wiki/Vertex_cut" class="mw-redirect" title="Vertex cut">vertex cut</a> which disconnects the graph is the complete set of vertices. The <a href="/wiki/Complement_graph" title="Complement graph">complement graph</a> of a complete graph is an <a href="/wiki/Empty_graph" class="mw-redirect" title="Empty graph">empty graph</a>. </p><p>If the edges of a complete graph are each given an <a href="/wiki/Orientation_(graph_theory)" title="Orientation (graph theory)">orientation</a>, the resulting <a href="/wiki/Directed_graph" title="Directed graph">directed graph</a> is called a <a href="/wiki/Tournament_(graph_theory)" title="Tournament (graph theory)">tournament</a>. </p><p><span class="texhtml mvar" style="font-style:italic;">K<sub>n</sub></span> can be decomposed into <span class="texhtml mvar" style="font-style:italic;">n</span> trees <span class="texhtml mvar" style="font-style:italic;">T<sub>i</sub></span> such that <span class="texhtml mvar" style="font-style:italic;">T<sub>i</sub></span> has <span class="texhtml mvar" style="font-style:italic;">i</span> vertices.<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> Ringel's conjecture asks if the complete graph <span class="texhtml"><i>K</i><sub>2<i>n</i>+1</sub></span> can be decomposed into copies of any tree with <span class="texhtml mvar" style="font-style:italic;">n</span> edges.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This is known to be true for sufficiently large <span class="texhtml mvar" style="font-style:italic;">n</span>.<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><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> </p><p>The number of all distinct <a href="/wiki/Path_(graph_theory)" title="Path (graph theory)">paths</a> between a specific pair of vertices in <span class="texhtml"><i>K</i><sub><i>n</i>+2</sub></span> is given<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> by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{n+2}=n!e_{n}=\lfloor en!\rfloor ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo>!</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>e</mi> <mi>n</mi> <mo>!</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{n+2}=n!e_{n}=\lfloor en!\rfloor ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5d8406df17e9b06c78e3ac732b4833a18f29fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.36ex; height:2.843ex;" alt="{\displaystyle w_{n+2}=n!e_{n}=\lfloor en!\rfloor ,}" /></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">e</span> refers to <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">Euler's constant</a>, and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{n}=\sum _{k=0}^{n}{\frac {1}{k!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{n}=\sum _{k=0}^{n}{\frac {1}{k!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9891d46e974a9f93178f83a87877ae00f34e06a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.484ex; height:7.009ex;" alt="{\displaystyle e_{n}=\sum _{k=0}^{n}{\frac {1}{k!}}.}" /></span></dd></dl> <p>The number of <a href="/wiki/Matching_(graph_theory)" title="Matching (graph theory)">matchings</a> of the complete graphs are given by the <a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">telephone numbers</a> </p> <dl><dd>1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, ... (sequence <span class="nowrap external"><a href="//oeis.org/A000085" class="extiw" title="oeis:A000085">A000085</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <p>These numbers give the largest possible value of the <a href="/wiki/Hosoya_index" title="Hosoya index">Hosoya index</a> for an <span class="texhtml mvar" style="font-style:italic;">n</span>-vertex graph.<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> The number of <a href="/wiki/Perfect_matching" title="Perfect matching">perfect matchings</a> of the complete graph <span class="texhtml mvar" style="font-style:italic;">K<sub>n</sub></span> (with <span class="texhtml mvar" style="font-style:italic;">n</span> even) is given by the <a href="/wiki/Double_factorial" title="Double factorial">double factorial</a> <span class="texhtml">(<i>n</i> – 1)!!</span>.<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> </p><p>The <a href="/wiki/Crossing_number_(graph_theory)" title="Crossing number (graph theory)">crossing numbers</a> up to <span class="texhtml"><i>K</i><sub>27</sub></span> are known, with <span class="texhtml"><i>K</i><sub>28</sub></span> requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Rectilinear Crossing numbers for <span class="texhtml mvar" style="font-style:italic;">K<sub>n</sub></span> are </p> <dl><dd>0, 0, 0, 0, 1, 3, 9, 19, 36, 62, 102, 153, 229, 324, 447, 603, 798, 1029, 1318, 1657, 2055, 2528, 3077, 3699, 4430, 5250, 6180, ... (sequence <span class="nowrap external"><a href="//oeis.org/A014540" class="extiw" title="oeis:A014540">A014540</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Geometry_and_topology">Geometry and topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complete_graph&action=edit&section=2" title="Edit section: Geometry and topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Csaszar_polyhedron_3D_model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Csaszar_polyhedron_3D_model.svg/100px-Csaszar_polyhedron_3D_model.svg.png" decoding="async" width="100" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Csaszar_polyhedron_3D_model.svg/150px-Csaszar_polyhedron_3D_model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Csaszar_polyhedron_3D_model.svg/200px-Csaszar_polyhedron_3D_model.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>Interactive <a href="/wiki/Csaszar_polyhedron" class="mw-redirect" title="Csaszar polyhedron">Csaszar polyhedron</a> model with vertices representing nodes. In <a class="external text" href="https://upload.wikimedia.org/wikipedia/commons/d/db/Csaszar_polyhedron_3D_model.svg">the SVG image</a>, move the mouse to rotate it.<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></figcaption></figure> <p>A complete graph with <span class="texhtml mvar" style="font-style:italic;">n</span> nodes represents the edges of an <span class="texhtml">(<i>n</i> – 1)</span>-<a href="/wiki/Simplex" title="Simplex">simplex</a>. Geometrically <span class="texhtml"><i>K</i><sub>3</sub></span> forms the edge set of a <a href="/wiki/Triangle" title="Triangle">triangle</a>, <span class="texhtml"><i>K</i><sub>4</sub></span> a <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>, etc. The <a href="/wiki/Cs%C3%A1sz%C3%A1r_polyhedron" title="Császár polyhedron">Császár polyhedron</a>, a nonconvex polyhedron with the topology of a <a href="/wiki/Torus" title="Torus">torus</a>, has the complete graph <span class="texhtml"><i>K</i><sub>7</sub></span> as its <a href="/wiki/Skeleton_(topology)" class="mw-redirect" title="Skeleton (topology)">skeleton</a>.<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> Every <a href="/wiki/Neighborly_polytope" title="Neighborly polytope">neighborly polytope</a> in four or more dimensions also has a complete skeleton. </p><p><span class="texhtml"><i>K</i><sub>1</sub></span> through <span class="texhtml"><i>K</i><sub>4</sub></span> are all <a href="/wiki/Planar_graph" title="Planar graph">planar graphs</a>. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph <span class="texhtml"><i>K</i><sub>5</sub></span> plays a key role in the characterizations of planar graphs: by <a href="/wiki/Kuratowski%27s_theorem" title="Kuratowski's theorem">Kuratowski's theorem</a>, a graph is planar if and only if it contains neither <span class="texhtml"><i>K</i><sub>5</sub></span> nor the <a href="/wiki/Complete_bipartite_graph" title="Complete bipartite graph">complete bipartite graph</a> <span class="texhtml"><i>K</i><sub>3,3</sub></span> as a subdivision, and by <a href="/wiki/Wagner%27s_theorem" title="Wagner's theorem">Wagner's theorem</a> the same result holds for <a href="/wiki/Graph_minor" title="Graph minor">graph minors</a> in place of subdivisions. As part of the <a href="/wiki/Petersen_family" title="Petersen family">Petersen family</a>, <span class="texhtml"><i>K</i><sub>6</sub></span> plays a similar role as one of the <a href="/wiki/Forbidden_minor" class="mw-redirect" title="Forbidden minor">forbidden minors</a> for <a href="/wiki/Linkless_embedding" title="Linkless embedding">linkless embedding</a>.<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> In other words, and as Conway and Gordon<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> proved, every embedding of <span class="texhtml"><i>K</i><sub>6</sub></span> into three-dimensional space is intrinsically linked, with at least one pair of linked triangles. Conway and Gordon also showed that any three-dimensional embedding of <span class="texhtml"><i>K</i><sub>7</sub></span> contains a <a href="/wiki/Hamiltonian_cycle" class="mw-redirect" title="Hamiltonian cycle">Hamiltonian cycle</a> that is embedded in space as a <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">nontrivial knot</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complete_graph&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Complete graphs on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> vertices, for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> between 1 and 12, are shown below along with the numbers of edges: </p> <table class="wikitable"> <tbody><tr> <th><span class="texhtml"><i>K</i><sub>1</sub>: 0</span> </th> <th><span class="texhtml"><i>K</i><sub>2</sub>: 1</span> </th> <th><span class="texhtml"><i>K</i><sub>3</sub>: 3</span> </th> <th><span class="texhtml"><i>K</i><sub>4</sub>: 6</span> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Complete_graph_K1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Complete_graph_K1.svg/140px-Complete_graph_K1.svg.png" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Complete_graph_K1.svg/210px-Complete_graph_K1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Complete_graph_K1.svg/280px-Complete_graph_K1.svg.png 2x" data-file-width="10000" data-file-height="10000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Complete_graph_K2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Complete_graph_K2.svg/140px-Complete_graph_K2.svg.png" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Complete_graph_K2.svg/210px-Complete_graph_K2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Complete_graph_K2.svg/280px-Complete_graph_K2.svg.png 2x" data-file-width="10000" data-file-height="10000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Complete_graph_K3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Complete_graph_K3.svg/140px-Complete_graph_K3.svg.png" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Complete_graph_K3.svg/210px-Complete_graph_K3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Complete_graph_K3.svg/280px-Complete_graph_K3.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:3-simplex_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/3-simplex_graph.svg/140px-3-simplex_graph.svg.png" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/3-simplex_graph.svg/210px-3-simplex_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/3-simplex_graph.svg/280px-3-simplex_graph.svg.png 2x" data-file-width="185" data-file-height="185" /></a></span> </td></tr> <tr> <th><span class="texhtml"><i>K</i><sub>5</sub>: 10</span> </th> <th><span class="texhtml"><i>K</i><sub>6</sub>: 15</span> </th> <th><span class="texhtml"><i>K</i><sub>7</sub>: 21</span> </th> <th><span class="texhtml"><i>K</i><sub>8</sub>: 28</span> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:4-simplex_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/4-simplex_graph.svg/140px-4-simplex_graph.svg.png" decoding="async" width="140" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/4-simplex_graph.svg/210px-4-simplex_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/4-simplex_graph.svg/280px-4-simplex_graph.svg.png 2x" data-file-width="188" data-file-height="195" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:5-simplex_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/5-simplex_graph.svg/140px-5-simplex_graph.svg.png" decoding="async" width="140" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/5-simplex_graph.svg/210px-5-simplex_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/5-simplex_graph.svg/280px-5-simplex_graph.svg.png 2x" data-file-width="223" data-file-height="195" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:6-simplex_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/6-simplex_graph.svg/140px-6-simplex_graph.svg.png" decoding="async" width="140" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/6-simplex_graph.svg/210px-6-simplex_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/6-simplex_graph.svg/280px-6-simplex_graph.svg.png 2x" data-file-width="228" data-file-height="233" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:7-simplex_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/7-simplex_graph.svg/140px-7-simplex_graph.svg.png" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/7-simplex_graph.svg/210px-7-simplex_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/7-simplex_graph.svg/280px-7-simplex_graph.svg.png 2x" data-file-width="253" data-file-height="253" /></a></span> </td></tr> <tr> <th><span class="texhtml"><i>K</i><sub>9</sub>: 36</span> </th> <th><span class="texhtml"><i>K</i><sub>10</sub>: 45</span> </th> <th><span class="texhtml"><i>K</i><sub>11</sub>: 55</span> </th> <th><span class="texhtml"><i>K</i><sub>12</sub>: 66</span> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:8-simplex_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/8-simplex_graph.svg/140px-8-simplex_graph.svg.png" decoding="async" width="140" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/8-simplex_graph.svg/210px-8-simplex_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/8-simplex_graph.svg/280px-8-simplex_graph.svg.png 2x" data-file-width="260" data-file-height="263" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:9-simplex_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/9-simplex_graph.svg/140px-9-simplex_graph.svg.png" decoding="async" width="140" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/9-simplex_graph.svg/210px-9-simplex_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/9-simplex_graph.svg/280px-9-simplex_graph.svg.png 2x" data-file-width="280" data-file-height="268" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:10-simplex_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/10-simplex_graph.svg/140px-10-simplex_graph.svg.png" decoding="async" width="140" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/10-simplex_graph.svg/210px-10-simplex_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/10-simplex_graph.svg/280px-10-simplex_graph.svg.png 2x" data-file-width="286" data-file-height="290" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:11-simplex_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/11-simplex_graph.svg/140px-11-simplex_graph.svg.png" decoding="async" width="140" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/11-simplex_graph.svg/210px-11-simplex_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/11-simplex_graph.svg/280px-11-simplex_graph.svg.png 2x" data-file-width="305" data-file-height="305" /></a></span> </td></tr></tbody></table> <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=Complete_graph&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/Network_topology#Decentralization" title="Network topology">Fully connected network</a>, in computer networking</li> <li><a href="/wiki/Complete_bipartite_graph" title="Complete bipartite graph">Complete bipartite graph</a> (or <b>biclique</b>), a special <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite graph</a> where every vertex on one side of the bipartition is connected to every vertex on the other side</li> <li>The <a href="/wiki/Simplex" title="Simplex">simplex</a>, which is identical to a <b>complete graph</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" /></span> vertices, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> is the <a href="/wiki/Dimension" title="Dimension">dimension</a> of the simplex.</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=Complete_graph&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"><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="CITEREFBang-JensenGutin2018" class="citation cs2">Bang-Jensen, Jørgen; Gutin, Gregory (2018), "Basic Terminology, Notation and Results", in Bang-Jensen, Jørgen; Gutin, Gregory (eds.), <i>Classes of Directed Graphs</i>, Springer Monographs in Mathematics, Springer International Publishing, pp. <span class="nowrap">1–</span>34, <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%2F978-3-319-71840-8_1">10.1007/978-3-319-71840-8_1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-71839-2" title="Special:BookSources/978-3-319-71839-2"><bdi>978-3-319-71839-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Basic+Terminology%2C+Notation+and+Results&rft.btitle=Classes+of+Directed+Graphs&rft.series=Springer+Monographs+in+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E34&rft.pub=Springer+International+Publishing&rft.date=2018&rft_id=info%3Adoi%2F10.1007%2F978-3-319-71840-8_1&rft.isbn=978-3-319-71839-2&rft.aulast=Bang-Jensen&rft.aufirst=J%C3%B8rgen&rft.au=Gutin%2C+Gregory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+graph" class="Z3988"></span>; see <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IHJgDwAAQBAJ&pg=PA17">p. 17</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKnuth2013" class="citation cs2"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (2013), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vj1oAgAAQBAJ&pg=PA7">"Two thousand years of combinatorics"</a>, in Wilson, Robin; Watkins, John J. 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Schneider">Schneider, Fred B.</a> (1993), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZWTDQ6H6gsUC&pg=PA436"><i>A Logical Approach to Discrete Math</i></a>, Springer-Verlag, p. 436, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0387941150" title="Special:BookSources/0387941150"><bdi>0387941150</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Logical+Approach+to+Discrete+Math&rft.pages=436&rft.pub=Springer-Verlag&rft.date=1993&rft.isbn=0387941150&rft.aulast=Gries&rft.aufirst=David&rft.au=Schneider%2C+Fred+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZWTDQ6H6gsUC%26pg%3DPA436&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+graph" class="Z3988"></span>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPirnot2000" class="citation cs2">Pirnot, Thomas L. 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Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CompleteGraph.html">"Complete 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=Complete+Graph&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCompleteGraph.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+graph" class="Z3988"></span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline 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