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<!doctype html> <html lang="en" class="discretemathematics historyandterminology"> <head> <title>Chromatic Polynomial -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Chromatic Polynomial" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="The chromatic polynomial pi_G(z) of an undirected graph G, also denoted C(G;z) (Biggs 1973, p. 106) and P(G,x) (Godsil and Royle 2001, p. 358), is a polynomial which encodes the number of distinct ways to color the vertices of G (where colorings are counted as distinct even if they differ only by permutation of colors). For a graph G on n vertices that can be colored in k_0=0 ways with no colors, k_1 way with one color, ..., and k_n ways with n colors, the chromatic polynomial of G is..." /> <meta name="description" content="The chromatic polynomial pi_G(z) of an undirected graph G, also denoted C(G;z) (Biggs 1973, p. 106) and P(G,x) (Godsil and Royle 2001, p. 358), is a polynomial which encodes the number of distinct ways to color the vertices of G (where colorings are counted as distinct even if they differ only by permutation of colors). For a graph G on n vertices that can be colored in k_0=0 ways with no colors, k_1 way with one color, ..., and k_n ways with n colors, the chromatic polynomial of G is..." /> <meta name="DC.Date.Created" scheme="W3CDTF" content="2000-03-04" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2007-05-01" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2008-03-01" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2008-07-10" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2008-07-20" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2008-08-17" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2008-10-06" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2008-10-13" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2011-07-19" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2013-12-05" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-05-05" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-05-31" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-06-04" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2015-09-28" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2017-04-02" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2021-07-18" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2024-01-22" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2024-07-26" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Discrete Mathematics:Graph Theory:Graph Coloring" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Discrete Mathematics:Graph Theory:Graph Properties:Graph Polynomials" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:History and Terminology:Wolfram Language Commands" /> <meta name="DC.Subject" scheme="MSC_2000" content="05C15" /> <meta name="DC.Rights" content="Copyright 1999-2025 Wolfram Research, Inc. 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For a graph G on n vertices that can be colored in k_0=0 ways with no colors, k_1 way with one color, ..., and k_n ways with n colors, the chromatic polynomial of G is..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Chromatic Polynomial -- from Wolfram MathWorld"> <meta name="twitter:description" content="The chromatic polynomial pi_G(z) of an undirected graph G, also denoted C(G;z) (Biggs 1973, p. 106) and P(G,x) (Godsil and Royle 2001, p. 358), is a polynomial which encodes the number of distinct ways to color the vertices of G (where colorings are counted as distinct even if they differ only by permutation of colors). For a graph G on n vertices that can be colored in k_0=0 ways with no colors, k_1 way with one color, ..., and k_n ways with n colors, the chromatic polynomial of G is..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_ChromaticPolynomial.png"> <link rel="canonical" href="https://mathworld.wolfram.com/ChromaticPolynomial.html" /> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <script async src="/common/javascript/analytics.js"></script> <script async src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script async src="/common/javascript/wal/latest/walLoad.js"></script> <link rel="stylesheet" href="/css/styles.css"> <link rel="preload" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css" as="style" onload="this.onload=null;this.rel='stylesheet'"> <noscript><link rel="stylesheet" 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href="/topics/GraphTheory.html">Graph Theory</a> </li> <li> <a href="/topics/GraphColoring.html">Graph Coloring</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/GraphTheory.html">Graph Theory</a> </li> <li> <a href="/topics/GraphProperties.html">Graph Properties</a> </li> <li> <a href="/topics/GraphPolynomials.html">Graph Polynomials</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/WolframLanguageCommands.html">Wolfram Language Commands</a> </li> </ul></nav>   <h1>Chromatic Polynomial</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/GraphTheory/ChromaticPolynomial.nb" download="ChromaticPolynomial.nb"><img src="/images/entries/download-notebook-icon.png" width="26" height="27" alt="DOWNLOAD Mathematica Notebook" /><span>Download <span class="display-i display-n__600">Wolfram </span>Notebook</span></a> </div>  <div class="entry-content"> <p> The chromatic polynomial <img src="/images/equations/ChromaticPolynomial/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="41" height="22" alt="pi_G(z)" /> of an <a href="/UndirectedGraph.html">undirected graph</a> <img src="/images/equations/ChromaticPolynomial/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />, also denoted <img src="/images/equations/ChromaticPolynomial/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="57" height="21" alt="C(G;z)" /> (Biggs 1973, p. 106) and <img src="/images/equations/ChromaticPolynomial/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="58" height="21" alt="P(G,x)" /> (Godsil and Royle 2001, p. 358), is a <a href="/Polynomial.html">polynomial</a> which encodes the number of distinct ways to color the vertices of <img src="/images/equations/ChromaticPolynomial/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> (where colorings are counted as distinct even if they differ only by permutation of colors). For a graph <img src="/images/equations/ChromaticPolynomial/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> on <img src="/images/equations/ChromaticPolynomial/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> vertices that can be colored in <img src="/images/equations/ChromaticPolynomial/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="22" alt="k_0=0" /> ways with no colors, <img src="/images/equations/ChromaticPolynomial/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="15" height="21" alt="k_1" /> way with one color, ..., and <img src="/images/equations/ChromaticPolynomial/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="15" height="21" alt="k_n" /> ways with <img src="/images/equations/ChromaticPolynomial/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> colors, the chromatic polynomial of <img src="/images/equations/ChromaticPolynomial/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is defined as the unique <a href="/LagrangeInterpolatingPolynomial.html">Lagrange interpolating polynomial</a> of degree <img src="/images/equations/ChromaticPolynomial/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> through the <img src="/images/equations/ChromaticPolynomial/Inline14.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n+1" /> points <img src="/images/equations/ChromaticPolynomial/Inline15.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="22" alt="(0,k_0)" />, <img src="/images/equations/ChromaticPolynomial/Inline16.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="21" alt="(1,k_1)" />, ..., <img src="/images/equations/ChromaticPolynomial/Inline17.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="21" alt="(n,k_n)" />. Evaluating the chromatic polynomial in variables <img src="/images/equations/ChromaticPolynomial/Inline18.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="z" /> at the points <img src="/images/equations/ChromaticPolynomial/Inline19.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="z=1" />, 2, ..., <img src="/images/equations/ChromaticPolynomial/Inline20.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> then recovers the numbers of 1-, 2-, ..., and <img src="/images/equations/ChromaticPolynomial/Inline21.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />-colorings. In fact, evaluating <img src="/images/equations/ChromaticPolynomial/Inline22.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="41" height="22" alt="pi_G(z)" /> at integers <img src="/images/equations/ChromaticPolynomial/Inline23.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="k>n" /> still gives the numbers of <img src="/images/equations/ChromaticPolynomial/Inline24.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" />-colorings. </p> <p> The chromatic polynomial is called the "chromial" for short by Bari (1974). </p> <p> The <a href="/ChromaticNumber.html">chromatic number</a> of a graph gives the smallest number of colors with which a graph can be colored, which is therefore the smallest positive integer <img src="/images/equations/ChromaticPolynomial/Inline25.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="z" /> such that <img src="/images/equations/ChromaticPolynomial/Inline26.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="69" height="22" alt="pi_G(z)>0" /> (Skiena 1990, p. 211). </p> <p> For example, the <a href="/CubicalGraph.html">cubical graph</a> <img src="/images/equations/ChromaticPolynomial/Inline27.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="22" alt="Q_3" /> has 1-, 2-, ... <a href="/k-Coloring.html"><i>k</i>-coloring</a> counts of 0, 2, 114, 2652, 29660, 198030, 932862, 3440024, ... (OEIS <a href="http://oeis.org/A140986">A140986</a>), resulting in chromatic polynomial </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/ChromaticPolynomial/NumberedEquation1.svg" data-src-small="/images/equations/ChromaticPolynomial/NumberedEquation1_400.svg" data-src-default="/images/equations/ChromaticPolynomial/NumberedEquation1.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="471" data-big="471 23" data-small="261 48" border="0" alt=" pi_(Q_3)(z)=z^8-12z^7+66z^6-214z^5+441z^4-572z^3+423z^2-133z. " /></td><td align="right" width="3"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr> </table> </div> <p> Evaluating <img src="/images/equations/ChromaticPolynomial/Inline28.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="47" height="24" alt="pi_(Q_3)(z)" /> at <img src="/images/equations/ChromaticPolynomial/Inline29.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="z=1" />, 2, ... then gives 0, 2, 114, 2652, 29660, 198030, 932862, 3440024, ... as expected. </p> <p> A <a href="/Root.html">root</a> of a chromatic polynomial is known as a <a href="/ChromaticRoot.html">chromatic root</a> and an <a href="/Interval.html">interval</a> containing no <a href="/ChromaticRoot.html">chromatic root</a> is called a <a href="/ChromaticRoot-FreeInterval.html">chromatic root-free interval</a>. </p> <p> The chromatic polynomial of a graph <img src="/images/equations/ChromaticPolynomial/Inline30.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="21" alt="g" /> in the variable <img src="/images/equations/ChromaticPolynomial/Inline31.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="z" /> can be determined in the <a href="http://www.wolfram.com/language/">Wolfram Language</a> using <tt><a href="http://reference.wolfram.com/language/ref/ChromaticPolynomial.html">ChromaticPolynomial</a></tt>[<i>g</i>, <i>x</i>]. Precomputed chromatic polynomials for many named graphs can be obtained using <tt><a href="http://reference.wolfram.com/language/ref/GraphData.html">GraphData</a></tt>[<i>graph</i>, <tt>"ChromaticPolynomial"</tt>][<i>z</i>]. </p> <p> The chromatic polynomial is multiplicative over graph components, so for a graph <img src="/images/equations/ChromaticPolynomial/Inline32.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> having connected components <img src="/images/equations/ChromaticPolynomial/Inline33.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="G_1" />, <img src="/images/equations/ChromaticPolynomial/Inline34.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="G_2" />, ..., the chromatic polynomial of <img src="/images/equations/ChromaticPolynomial/Inline35.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> itself is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/ChromaticPolynomial/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="113" height="23" alt=" pi_G=pi_(G_1)pi_(G_2).... " /></td><td align="right" width="3"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> The chromatic polynomial for a <a href="/Forest.html">forest</a> on <img src="/images/equations/ChromaticPolynomial/Inline36.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> vertices, <img src="/images/equations/ChromaticPolynomial/Inline37.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="m" /> edges, and with <img src="/images/equations/ChromaticPolynomial/Inline38.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="c" /> connected components is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/ChromaticPolynomial/NumberedEquation3.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="155" height="20" alt=" pi=(-1)^(n-c)x^c(1-x)^m. " /></td><td align="right" width="3"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr> </table> </div> <p> For a graph with <a href="/VertexCount.html">vertex count</a> <img src="/images/equations/ChromaticPolynomial/Inline39.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> and <img src="/images/equations/ChromaticPolynomial/Inline40.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="c" /> connected components, the chromatic polynomial <img src="/images/equations/ChromaticPolynomial/Inline41.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="pi(x)" /> is related to the <a href="/RankPolynomial.html">rank polynomial</a> <img src="/images/equations/ChromaticPolynomial/Inline42.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="53" height="21" alt="R(x,y)" /> and <a href="/TuttePolynomial.html">Tutte polynomial</a> <img src="/images/equations/ChromaticPolynomial/Inline43.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="T(x,y)" /> by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/ChromaticPolynomial/Inline44.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="29" height="20" alt="pi(x)" /></td><td align="center" width="14"><img src="/images/equations/ChromaticPolynomial/Inline45.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline46.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="104" height="23" alt="x^nR(-x^(-1),-1)" /></td><td align="right" width="10"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/ChromaticPolynomial/Inline47.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/ChromaticPolynomial/Inline48.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline49.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="144" height="20" alt="(-1)^(n-c)x^cT(1-x,0)" /></td><td align="right" width="10"> <div id="eqn5" class="eqnum"> (5) </div> </td></tr> </table> </div> <p> (extending Biggs 1993, p. 106). The chromatic polynomial of a <a href="/PlanarGraph.html">planar graph</a> <img src="/images/equations/ChromaticPolynomial/Inline50.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is related to the <a href="/FlowPolynomial.html">flow polynomial</a> <img src="/images/equations/ChromaticPolynomial/Inline51.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="23" alt="C_G^*(u)" /> of its <a href="/DualGraph.html">dual graph</a> <img src="/images/equations/ChromaticPolynomial/Inline52.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="G^*" /> by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/ChromaticPolynomial/NumberedEquation4.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="122" height="23" alt=" pi_G(x)=xC_(G^*)^*(x). " /></td><td align="right" width="3"> <div id="eqn6" class="eqnum"> (6) </div> </td></tr> </table> </div> <p> Chromatic polynomials are not diagnostic for graph isomorphism, i.e., two nonisomorphic graphs may share the same chromatic polynomial. A graph that is determined by its chromatic polynomial is said to be a <a href="/ChromaticallyUniqueGraph.html">chromatically unique graph</a>; nonisomorphic graphs sharing the same chromatic polynomial are said to be <a href="/ChromaticallyEquivalentGraphs.html">chromatically equivalent</a>. </p> <p> Chromatic polynomials of the <a href="/LadderGraph.html">ladder graph</a> <img src="/images/equations/ChromaticPolynomial/Inline53.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="P_2 square P_n" /> and <a href="/GridGraph.html">grid graph</a> <img src="/images/equations/ChromaticPolynomial/Inline54.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="P_2 square P_n" /> are considered by Yadav <i>et al. </i>(2024). The following table summarizes the chromatic polynomials for some simple graphs. Here <img src="/images/equations/ChromaticPolynomial/Inline55.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="25" height="22" alt="(z)_n" /> is the <a href="/FallingFactorial.html">falling factorial</a>. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left">graph</td><td align="left">chromatic polynomial</td></tr><tr style=""><td align="left"><a href="/BarbellGraph.html">barbell graph</a></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline56.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="55" height="31" alt="((z)_n^2(z-1))/z" /></td></tr><tr style=""><td align="left"><a href="/BookGraph.html">book graph</a> <img src="/images/equations/ChromaticPolynomial/Inline57.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="65" height="22" alt="S_(n+1) square P_2" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline58.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="162" height="25" alt="(z-1)z(z^2-3z+3)^n" /></td></tr><tr style=""><td align="left"><a href="/CentipedeGraph.html">centipede graph</a></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline59.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="88" height="21" alt="(z-1)^(2n-1)z" /></td></tr><tr style=""><td align="left"><a href="/CompleteGraph.html">complete graph</a> <img src="/images/equations/ChromaticPolynomial/Inline60.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="K_n" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline61.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="25" height="22" alt="(z)_n" /></td></tr><tr style=""><td align="left"><a href="/CycleGraph.html">cycle graph</a> <img src="/images/equations/ChromaticPolynomial/Inline62.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_n" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline63.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="160" height="21" alt="(-1)^n(z-1)+(z-1)^n" /></td></tr><tr style=""><td align="left"><a href="/GearGraph.html">gear graph</a></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline64.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="179" height="25" alt="z[z-2+(3-3z+z^2)^n]" /></td></tr><tr style=""><td align="left"><a href="/HelmGraph.html">helm graph</a></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline65.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="256" height="21" alt="z[(1-z)^n(z-2)+(z-2)^n(z-1)^n]" /></td></tr><tr style=""><td align="left"><a href="/LadderGraph.html">ladder graph</a> <img src="/images/equations/ChromaticPolynomial/Inline66.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="P_2 square P_n" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline67.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="176" height="28" alt="(z-1)z(z^2-3z+3)^(n-1)" /></td></tr><tr style=""><td align="left"><a href="/LadderRungGraph.html">ladder rung graph</a> <img src="/images/equations/ChromaticPolynomial/Inline68.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="nP_2" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline69.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="72" height="21" alt="z^n(z-1)^n" /></td></tr><tr style=""><td align="left"><a href="/MoebiusLadder.html">Möbius ladder</a> <img src="/images/equations/ChromaticPolynomial/Inline70.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="M_n" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline71.svg" data-src-small="/images/equations/ChromaticPolynomial/Inline71_400.svg" data-src-default="/images/equations/ChromaticPolynomial/Inline71.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="477" height="21" data-big="477 21" data-small="308 47" border="0" alt="-1+(1-z)^n-(3-z)^n+(-(1-z)^n+(3-z)^n)z+(3+(-3+z)z)^n" /></td></tr><tr style=""><td align="left"><a href="/PanGraph.html">pan graph</a></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline72.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="181" height="21" alt="(z-1)^(n+1)+(-1)^n(z-1)^2" /></td></tr><tr style=""><td align="left"><a href="/PathGraph.html">path graph</a> <img src="/images/equations/ChromaticPolynomial/Inline73.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="P_n" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline74.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="79" height="21" alt="z(z-1)^(n-1)" /></td></tr><tr style=""><td align="left"><a href="/PrismGraph.html">prism graph</a> <img src="/images/equations/ChromaticPolynomial/Inline75.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="21" alt="Y_n" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline76.svg" data-src-small="/images/equations/ChromaticPolynomial/Inline76_400.svg" data-src-default="/images/equations/ChromaticPolynomial/Inline76.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="500" height="21" data-big="500 21" data-small="345 47" border="0" alt="1+[z(z-3)+3]^n+z[(1-z)^n+(3-z)^n+z-3]-(1-z)^n-(3-z)^n" /></td></tr><tr style=""><td align="left"><a href="/StarGraph.html">star graph</a> <img src="/images/equations/ChromaticPolynomial/Inline77.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="S_n" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline78.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="79" height="21" alt="z(z-1)^(n-1)" /></td></tr><tr style=""><td align="left"><a href="/SunGraph.html">sun graph</a></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline79.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="82" height="22" alt="(z)_n(z-2)^n" /></td></tr><tr style=""><td align="left"><a href="/SunletGraph.html">sunlet graph</a> <img src="/images/equations/ChromaticPolynomial/Inline80.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="59" height="21" alt="C_n circledot K_1" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline81.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="148" height="21" alt="(z-1)^(2n)-(1-z)^(n-1)" /></td></tr><tr style=""><td align="left">triangular honeycomb rook graph</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline82.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="82" height="23" alt="product_(k=1)^(n)[(z)_k]^n" /></td></tr><tr style=""><td align="left"><a href="/WebGraph.html">web graph</a></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline83.svg" data-src-small="/images/equations/ChromaticPolynomial/Inline83_400.svg" data-src-default="/images/equations/ChromaticPolynomial/Inline83.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="766" height="25" data-big="766 25" data-small="311 80" border="0" alt="z[(1-z)^n+(3-z)^n+z-3](z-1)^n+(z-1)^n-[-(z-3)(z-1)]^n-[-(z-1)^2]^n+[(z-1)((z-3)z+3)]^n" /></td></tr><tr style=""><td align="left"><a href="/WheelGraph.html">wheel graph</a> <img src="/images/equations/ChromaticPolynomial/Inline84.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="W_n" /></td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline85.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="198" height="25" alt="z[(z-2)^(n-1)-(-1)^n(z-2)]" /></td></tr> </table> </div> <p> The following table summarizes the recurrence relations for chromatic polynomials for some simple classes of graphs. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left">graph</td><td align="right">order</td><td align="left">recurrence</td></tr><tr style=""><td align="left"><a href="/AntiprismGraph.html">antiprism graph</a></td><td align="right">4</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline86.svg" data-src-small="/images/equations/ChromaticPolynomial/Inline86_400.svg" data-src-default="/images/equations/ChromaticPolynomial/Inline86.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="719" height="25" data-big="719 25" data-small="369 54" border="0" alt="p_n=(z^2-6z+10)p_(n-1)+(z-3)(2z^2-9z+11)p_(n-2)+(z^2-6z+10)(z-2)^2p_(n-3)-(z-2)^4p_(n-4)" /></td></tr><tr style=""><td align="left"><a href="/BarbellGraph.html">barbell graph</a></td><td align="right">1</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline87.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="156" height="22" alt="p_n=(z-n+1)^2p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/BookGraph.html">book graph</a> <img src="/images/equations/ChromaticPolynomial/Inline88.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="65" height="22" alt="S_(n+1) square P_2" /></td><td align="right">1</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline89.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="168" height="25" alt="p_n=(z^2-3z+3)p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/CentipedeGraph.html">centipede graph</a></td><td align="right">1</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline90.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="128" height="22" alt="p_n=(z-1)^2p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/CompleteGraph.html">complete graph</a> <img src="/images/equations/ChromaticPolynomial/Inline91.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="K_n" /></td><td align="right">1</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline92.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="149" height="22" alt="p_n=(z-n+1)p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/CycleGraph.html">cycle graph</a> <img src="/images/equations/ChromaticPolynomial/Inline93.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_n" /></td><td align="right">2</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline94.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="222" height="22" alt="p_n=(z-2)p_(n-1)+(z-1)p_(n-2)" /></td></tr><tr style=""><td align="left"><a href="/GearGraph.html">gear graph</a></td><td align="right">2</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline95.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="316" height="25" alt="p_n=(z^2-3z+4)p_(n-1)-(z^2-3z+3)p_(n-2)" /></td></tr><tr style=""><td align="left"><a href="/HelmGraph.html">helm graph</a></td><td align="right">2</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline96.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="329" height="22" alt="p_n=(z-3)(z-1)p_(n-1)+(z-2)(z-1)^2p_(n-2)" /></td></tr><tr style=""><td align="left"><a href="/LadderGraph.html">ladder graph</a> <img src="/images/equations/ChromaticPolynomial/Inline97.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="P_2 square P_n" /></td><td align="right">1</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline98.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="168" height="25" alt="p_n=(z^2-3z+3)p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/LadderRungGraph.html">ladder rung graph</a> <img src="/images/equations/ChromaticPolynomial/Inline99.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="nP_2" /></td><td align="right">1</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline100.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="133" height="22" alt="p_n=z(z-1)p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/MoebiusLadder.html">Möbius ladder</a></td><td align="right">4</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline101.svg" data-src-small="/images/equations/ChromaticPolynomial/Inline101_400.svg" data-src-default="/images/equations/ChromaticPolynomial/Inline101.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="956" height="25" data-big="956 25" data-small="383 83" border="0" alt="p_n=(8-5z+z^2)p_(n-1)+(-22+27z-12z^2+2z^3)p_(n-2)+(24-43z+29z^2-9z^3+z^4)p_(n-3)+(-9+21z-18z^2+7z^3-z^4)p_(n-4)" /></td></tr><tr style=""><td align="left"><a href="/PanGraph.html">pan graph</a></td><td align="right">2</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline102.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="222" height="22" alt="p_n=(z-1)p_(n-2)+(z-2)p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/PathGraph.html">path graph</a> <img src="/images/equations/ChromaticPolynomial/Inline103.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="P_n" /></td><td align="right">1</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline104.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="121" height="22" alt="p_n=(z-1)p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/PrismGraph.html">prism graph</a> <img src="/images/equations/ChromaticPolynomial/Inline105.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="21" alt="Y_n" /></td><td align="right">4</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline106.svg" data-src-small="/images/equations/ChromaticPolynomial/Inline106_400.svg" data-src-default="/images/equations/ChromaticPolynomial/Inline106.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="905" height="25" data-big="905 25" data-small="356 83" border="0" alt="p_n=(z^2-5z+8)p_(n-1)+(z-2)(2z^2-8z+11)p_(n-2)+(z^4-9z^3+29z^2-43z+24)p_(n-3)-(z-3)(z-1)(z^2-3z+3)p_(n-4)" /></td></tr><tr style=""><td align="left"><a href="/StarGraph.html">star graph</a> <img src="/images/equations/ChromaticPolynomial/Inline107.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="S_n" /></td><td align="right">1</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline108.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="121" height="22" alt="p_n=(z-1)p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/SunletGraph.html">sunlet graph</a> <img src="/images/equations/ChromaticPolynomial/Inline109.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="59" height="21" alt="C_n circledot K_1" /></td><td align="right">2</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline110.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="279" height="22" alt="p_n=(z-1)(z-2)p_(n-1)+(z-1)^3p_(n-2)" /></td></tr><tr style=""><td align="left"><a href="/WebGraph.html">web graph</a></td><td align="right">4</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline111.svg" data-src-small="/images/equations/ChromaticPolynomial/Inline111_400.svg" data-src-default="/images/equations/ChromaticPolynomial/Inline111.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="628" height="54" data-big="628 54" data-small="302 112" border="0" alt="p_n=p_n=(z^2-5z+8)(z-1)p_(n-1)+(z-2)(2z^2-8z+11)(z-1)^2p_(n-2)+(z^4-9z^3+29z^2-43z+24)(z-1)^3p_(n-3)-(z-3)(z^2-3z+3)(z-1)^5p_(n-4)" /></td></tr><tr style=""><td align="left"><a href="/WheelGraph.html">wheel graph</a> <img src="/images/equations/ChromaticPolynomial/Inline112.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="W_n" /></td><td align="right">2</td><td align="left"><img src="/images/equations/ChromaticPolynomial/Inline113.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="222" height="22" alt="p_n=(z-2)p_(n-2)+(z-3)p_(n-1)" /></td></tr> </table> </div> <p> The chromatic polynomial of a <a href="/DisconnectedGraph.html">disconnected graph</a> is the product of the chromatic polynomials of its <a href="/ConnectedComponent.html">connected components</a>. The chromatic polynomial of a graph of order <img src="/images/equations/ChromaticPolynomial/Inline114.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> has degree <img src="/images/equations/ChromaticPolynomial/Inline115.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />, with leading coefficient 1 and constant term 0. Furthermore, the coefficients alternate signs, and the coefficient of the <img src="/images/equations/ChromaticPolynomial/Inline116.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="47" height="21" alt="(n-1)" />st term is <img src="/images/equations/ChromaticPolynomial/Inline117.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="25" height="21" alt="-m" />, where <img src="/images/equations/ChromaticPolynomial/Inline118.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="m" /> is the number of edges. Interestingly, <img src="/images/equations/ChromaticPolynomial/Inline119.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="22" alt="pi_G(-1)" /> is equal to the number of acyclic orientations of <img src="/images/equations/ChromaticPolynomial/Inline120.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> (Stanley 1973). </p> <p> Except for special cases (such as <a href="/Tree.html">trees</a>), the calculation of <img src="/images/equations/ChromaticPolynomial/Inline121.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="41" height="22" alt="pi_G(z)" /> is exponential in the minimum number of edges in <img src="/images/equations/ChromaticPolynomial/Inline122.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> and the <a href="/GraphComplement.html">graph complement</a> <img src="/images/equations/ChromaticPolynomial/Inline123.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="14" height="21" alt="G^_" /> (Skiena 1990, p. 211), and calculating the chromatic polynomial of a <a href="/Graph.html">graph</a> is at least an <a href="/NP-CompleteProblem.html">NP-complete problem</a> (Skiena 1990, pp. 211-212). </p> <p> Tutte (1970) showed that the chromatic polynomial of a planar triangulation of a sphere possess a <a href="/Root.html">root</a> close to <img src="/images/equations/ChromaticPolynomial/Inline124.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="184" height="22" alt="phi^2=phi+1=2.618033..." /> (OEIS <a href="http://oeis.org/A104457">A104457</a>), where <img src="/images/equations/ChromaticPolynomial/Inline125.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="phi" /> is the <a href="/GoldenRatio.html">golden ratio</a>. More precisely, if <img src="/images/equations/ChromaticPolynomial/Inline126.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> is the number of <a href="/GraphVertex.html">graph vertices</a> of such a graph <img src="/images/equations/ChromaticPolynomial/Inline127.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />, then </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/ChromaticPolynomial/NumberedEquation5.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="94" height="23" alt=" pi_G(phi^2)<=phi^(5-n) " /></td><td align="right" width="3"> <div id="eqn7" class="eqnum"> (7) </div> </td></tr> </table> </div> <p> (Tutte 1970, Le Lionnais 1983). </p> <p> Read (1968) conjectured that, for any chromatic polynomial </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/ChromaticPolynomial/NumberedEquation6.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="158" height="20" alt=" pi(z)=c_nz^n+...+c_1z, " /></td><td align="right" width="3"> <div id="eqn8" class="eqnum"> (8) </div> </td></tr> </table> </div> <p> there does not exist a <img src="/images/equations/ChromaticPolynomial/Inline128.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="121" height="21" alt="1<=p<=q<=r<=n" /> such that <img src="/images/equations/ChromaticPolynomial/Inline129.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="67" height="25" alt="|c_p|>|c_q|" /> and <img src="/images/equations/ChromaticPolynomial/Inline130.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="64" height="25" alt="|c_q|<|c_r|" /> (Skiena 1990, p. 221). </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/ChromaticInterval.html">Chromatic Interval</a>, <a href="/ChromaticInvariant.html">Chromatic Invariant</a>, <a href="/ChromaticNumber.html">Chromatic Number</a>, <a href="/ChromaticRoot.html">Chromatic Root</a>, <a href="/ChromaticallyEquivalentGraphs.html">Chromatically Equivalent Graphs</a>, <a href="/ChromaticallyUniqueGraph.html">Chromatically Unique Graph</a>, <a href="/FlowPolynomial.html">Flow Polynomial</a>, <a href="/k-Coloring.html"><i>k</i>-Coloring</a>, <a href="/k-ChromaticGraph.html"><i>k</i>-Chromatic Graph</a>, <a href="/k-ColorableGraph.html"><i>k</i>-Colorable Graph</a>, <a href="/Q-ChromaticPolynomial.html">Q-Chromatic Polynomial</a>, <a href="/RankPolynomial.html">Rank Polynomial</a>, <a href="/SigmaPolynomial.html">Sigma Polynomial</a>, <a href="/TuttePolynomial.html">Tutte Polynomial</a>, <a href="/VertexColoring.html">Vertex Coloring</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=.999+with+123+repeating">.999 with 123 repeating</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=factor+sin+x+%2B+sin+y">factor sin x + sin y</a></li> <li><a target="_blank" href="http://www.wolframalpha.com/input/?i=g%28n%2B1%29%3Dn%5E2%2Bg%28n%29">g(n+1)=n^2+g(n)</a></li> </ul> </div> </div>   <h2>References</h2><cite>Bari, R. A. "Chromatically Equivalent Graphs." In <i>Graphs and Combinatorics</i> (Ed. R. A. Bari and F. Harary). Berlin: Springer-Verlag, pp. 186-200, 1974.</cite><cite>Berman, G. and Tutte, W. T. "The Golden Root of a Chromatic Polynomial." <i>J. Combin. Th.</i> <b>6</b>, 301-302, 1969.</cite><cite>Biggs, N. L. "Chromatic Polynomials and Spanning Trees." Ch. 14 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521458978/ref=nosim/ericstreasuretro">Algebraic Graph Theory, 2nd ed.</a></i> Cambridge, England: Cambridge University Press, pp. 106-111, 1993.</cite><cite>Birkhoff, G. D. "A Determinant Formula for the Number of Ways of Coloring a Map." <i>Ann. Math.</i> <b>14</b>, 42-46, 1912.</cite><cite>Birkhoff, G. D. and Lewis, D. C. "Chromatic Polynomials." <i>Trans. Amer. Math. Soc.</i> <b>60</b>, 355-451, 1946.</cite><cite>Chvátal, V. "A Note on Coefficients of Chromatic Polynomials." <i>J. Combin. Th.</i> <b>9</b>, 95-96, 1970.</cite><cite>Dong, F. M., Koh, K. M.; and Teo, K. L. <i><a href="http://www.amazon.com/exec/obidos/ASIN/9812563172/ref=nosim/ericstreasuretro">Chromatic Polynomials and Chromaticity of Graphs.</a></i> Singapore: World Scientific, 2005.</cite><cite>Erdős, P. and Hajnal, A. "On Chromatic Numbers of Graphs and Set-Systems." <i>Acta Math. Acad. Sci. Hungar.</i> <b>17</b>, 61-99, 1966.</cite><cite>Godsil, C. and Royle, G. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387952209/ref=nosim/ericstreasuretro">Algebraic Graph Theory.</a></i> New York: Springer-Verlag, 2001.</cite><cite>Le Lionnais, F. <i><a href="http://www.amazon.com/exec/obidos/ASIN/2705614079/ref=nosim/ericstreasuretro">Les nombres remarquables.</a></i> Paris: Hermann, p. 46, 1983.</cite><cite>Read, R. C. "An Introduction to Chromatic Polynomials." <i>J. Combin. Th.</i> <b>4</b>, 52-71, 1968.</cite><cite>Saaty, T. L. and Kainen, P. C. "Chromatic Numbers and Chromatic Polynomials." Ch. 6 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486650928/ref=nosim/ericstreasuretro">The Four-Color Problem: Assaults and Conquest.</a></i> New York: Dover, pp. 134-163 1986.</cite><cite>Skiena, S. "Chromatic Polynomials." §5.5.1 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521806860/ref=nosim/ericstreasuretro">Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica.</a></i> Reading, MA: Addison-Wesley, pp. 210-212, 1990.</cite><cite>Sloane, N. J. A. Sequences <a href="http://oeis.org/A104457">A104457</a> and <a href="http://oeis.org/A140986">A140986</a> in "The On-Line Encyclopedia of Integer Sequences."</cite><cite>Stanley, R. P. "Acyclic Orientations of Graphs." <i>Disc. Math.</i> <b>5</b>, 171-178, 1973.</cite><cite>Tutte, W. T. "On Chromatic Polynomials and the Golden Ratio." <i>J. Combin. Th.</i> <b>9</b>, 289-296, 1970.</cite><cite>Yadav, R.; Sehgal, A.; Sehgal, S.; and Malik, A. "The Chromatic Polynomial of Grid Graph <img src="/images/equations/ChromaticPolynomial/Inline131.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="22" alt="P_3 square P_n" />." <i>J. Appl. Math. Comput.</i>, 2024.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/chromatic_polynomial/vx/43/kx/" title="Chromatic Polynomial" target="_blank">Chromatic Polynomial</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Chromatic Polynomial." 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