Biclique Coverings and the Chromatic Number

Abstract

Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \medskip \noindent $\bullet$ If the bicliques partition the edges of $G$, then their number is at least $2^{\sqrt{{\log }_{2}k}}$. This is the first improvement of the easy lower bound of ${\log }_{2}k$, while the Alon-Saks-Seymour conjecture states that this can be improved to $k-1$. \medskip \noindent $\bullet$ The sum of the orders of the bicliques is at least $(1-o(1))k{\log }_{2}k$. This generalizes, in asymptotic form, a result of Katona and Szemerédi who proved that the minimum is $k{\log }_{2}k$ when $G$ is a clique.