On biclique partitions of the complete graph

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Abstract

A biclique partition of a graph G is an edge coloring of G such that the edge subgraph formed by the edges of any given color is a complete bipartite graph. A claw is a graph isomorphic to a K1,a for some a. A siamese claw is a graph isomorphic to K2,a for some a. We find necessary and sufficient conditions for Kn to be partitioned into claws K1,a1,…,Kl,al where l=n−1 or l=n. Let Xi(i=1,…,m) be a collection of two-element subsets of {1,…,n}. We study the problem of finding subsets Yi(i=1,…,m) of {1,…,n} such that the siamese claws {(a,b): aXi, bYi} (i=1,…,m) partition the edge set of Kn. It is proved that such Yi's exist if and only if there is a perfect matching in a graph G̃ associated with the graph G, the graph whose edges are the Xi's.

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Research partially supported by a University of Wyoming Basic Research Grant.