Regarding Two Conjectures on Clique and Biclique Partitions

Abstract

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures – one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.