On the p -Edge Clique over Nuber of Complete Bipartite Graphs
Michael S. Jacobson · SIAM Journal on Discrete Mathematics · 1992
Given a digraph $D = ( V,A )$ and a positive integer p, the p-competition graph of D, denoted $C_p ( D )$, is defined to have vertex set V and for $x,y \in V,xy \in E ( C_p ( D ) )$ if and only if there are at least p distinct vertices $v_1 ,v_2 , \cdots ,v_p \in V ( D )$ such that $xv_i $ and $yv_i \in A( D )$ for $i = 1,2, \cdots ,p$. This paper furthers the study of complete bipartite graphs that are p-competition graphs. The primary technique used is the concept of the p-edge clique cover (p-ECC) number. General results are given for $K_3 $-free graphs, as well as the result that $K_{n,n} $ is not a 2-competition graph for $n\geqq 4$.