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completeMultiPartite -- returns a complete multipartite graph

Synopsis

Description

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.

i1 : R = QQ[a,b,c,x,y,z];
 
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
 
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
 
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

When n is the number of variables and M = 1, we recover the complete graph.

i5 : R = QQ[a,b,c,d,e];
 
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
 
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
 
i8 : t1 == t2

o8 = true

See also

Ways to use completeMultiPartite :

For the programmer

The object completeMultiPartite is a method function.