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<head><title>completeMultiPartite -- returns a complete multipartite graph</title>
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<div><h1>completeMultiPartite -- returns a complete multipartite graph</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>K = completeMultiPartite(R,n,m)</tt><br/><tt>K = completeMultiPartite(R,L)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span></span></li>
<li><span><tt>n</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, number of partitions</span></li>
<li><span><tt>m</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, size of each partition</span></li>
<li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of integers giving the size of each partition, or a list of partitions that are lists of variables</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>K</tt>, <span>a <a href="___Graph.html">graph</a></span>, the complete multipartite graph on the given partitions</span></li>
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<div class="single"><h2>Description</h2>
<div><div>A complete multipartite graph is a graph with a partition of the vertices such that every pair of vertices, not both from the same partition, is an edge of the graph. The partitions can be specified by their number and size, by a list of sizes, or by an explicit partition of the variables. Not all variables of the ring need to be used.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[a,b,c,x,y,z];</pre>
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<tr><td><pre>i2 : completeMultiPartite(R,2,3)
o2 = Graph{edges => {{a, x}, {a, y}, {a, z}, {b, x}, {b, y}, {b, z}, {c, x}, {c, y}, {c, z}}}
ring => R
vertices => {a, b, c, x, y, z}
o2 : Graph</pre>
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<tr><td><pre>i3 : completeMultiPartite(R,{2,4})
o3 = Graph{edges => {{a, c}, {a, x}, {a, y}, {a, z}, {b, c}, {b, x}, {b, y}, {b, z}}}
ring => R
vertices => {a, b, c, x, y, z}
o3 : Graph</pre>
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<tr><td><pre>i4 : completeMultiPartite(R,{{a,b,c,x},{y,z}})
o4 = Graph{edges => {{a, y}, {a, z}, {b, y}, {b, z}, {c, y}, {c, z}, {x, y}, {x, z}}}
ring => R
vertices => {a, b, c, x, y, z}
o4 : Graph</pre>
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<div>When <tt>n</tt> is the number of variables and <tt>M = 1</tt>, we recover the complete graph.</div>
<table class="examples"><tr><td><pre>i5 : R = QQ[a,b,c,d,e];</pre>
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<tr><td><pre>i6 : t1 = completeMultiPartite(R,5,1)
o6 = Graph{edges => {{a, b}, {a, c}, {a, d}, {a, e}, {b, c}, {b, d}, {b, e}, {c, d}, {c, e}, {d, e}}}
ring => R
vertices => {a, b, c, d, e}
o6 : Graph</pre>
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<tr><td><pre>i7 : t2 = completeGraph R
o7 = Graph{edges => {{a, b}, {a, c}, {a, d}, {a, e}, {b, c}, {b, d}, {b, e}, {c, d}, {c, e}, {d, e}}}
ring => R
vertices => {a, b, c, d, e}
o7 : Graph</pre>
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<tr><td><pre>i8 : t1 == t2
o8 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_complete__Graph.html" title="returns a complete graph">completeGraph</a> -- returns a complete graph</span></li>
<li><span><a href="___Constructor_sp__Overview.html" title="a summary of the many ways of making graphs and hypergraphs">Constructor Overview</a> -- a summary of the many ways of making graphs and hypergraphs</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>completeMultiPartite</tt> :</h2>
<ul><li>completeMultiPartite(Ring,List)</li>
<li>completeMultiPartite(Ring,ZZ,ZZ)</li>
</ul>
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