On biclique partitions of the complete graph

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 K 1, a for some a. A siamese claw is a graph isomorphic to K 2, a for some a. We find necessary and sufficient conditions for K n to be partitioned into claws K 1, a 1 ,…, K l, a l where l= n−1 or l= n. Let X i ( i=1,…, m) be a collection of two-element subsets of {1,…, n}. We study the problem of finding subsets Y i ( i=1,…, m) of {1,…, n} such that the siamese claws {( a, b): a∈ X i , b∈ Y i } ( i=1,…, m) partition the edge set of K n . It is proved that such Y i '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 X i 's.