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<head><title>chromaticNumber -- computes the chromatic number of a hypergraph</title>
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<div><h1>chromaticNumber -- computes the chromatic number of a hypergraph</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>c = chromaticNumber H</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>H</tt>, <span>a <a href="___Hyper__Graph.html">hypergraph</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>c</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the chromatic number of <tt>H</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><div>Returns the chromatic number, the smallest number of colors needed to color the vertices of a graph. This method is based upon a result of Francisco-Ha-Van Tuyl that relates the chromatic number to an ideal membership problem.</div>
<table class="examples"><tr><td><pre>i1 : S = QQ[a..f];</pre>
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<tr><td><pre>i2 : c4 = cycle(S,4) -- 4-cycle; chromatic number = 2
o2 = Graph{edges => {{a, b}, {b, c}, {c, d}, {a, d}}}
ring => S
vertices => {a, b, c, d, e, f}
o2 : Graph</pre>
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<tr><td><pre>i3 : c5 = cycle(S,5) -- 5-cycle; chromatic number = 3
o3 = Graph{edges => {{a, b}, {b, c}, {c, d}, {d, e}, {a, e}}}
ring => S
vertices => {a, b, c, d, e, f}
o3 : Graph</pre>
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<tr><td><pre>i4 : k6 = completeGraph S -- complete graph on 6 vertices; chormatic number = 6
o4 = Graph{edges => {{a, b}, {a, c}, {a, d}, {a, e}, {a, f}, {b, c}, {b, d}, {b, e}, {b, f}, {c, d}, {c, e}, {c, f}, {d, e}, {d, f}, {e, f}}}
ring => S
vertices => {a, b, c, d, e, f}
o4 : Graph</pre>
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<tr><td><pre>i5 : chromaticNumber c4
o5 = 2</pre>
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<tr><td><pre>i6 : chromaticNumber c5
o6 = 3</pre>
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<tr><td><pre>i7 : chromaticNumber k6
o7 = 6</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>This method should not be used with a hypergraph that has an edge of cardinality one since no coloring is possible.</div>
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<div class="waystouse"><h2>Ways to use <tt>chromaticNumber</tt> :</h2>
<ul><li>chromaticNumber(HyperGraph)</li>
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