Biclique Coverings and the Chromatic Number

TL;DR

This paper investigates the minimum number and total size of bicliques needed to cover the edges of a graph with a given chromatic number, providing new lower bounds that improve upon previous results.

Contribution

It establishes new lower bounds on the number and total size of bicliques covering a graph's edges, advancing understanding of graph coverings related to chromatic number.

Findings

Number of bicliques partitioning edges is at least 2^{√log₂k}

Sum of biclique orders is at least (1-o(1))k log₂k

Improves previous bounds on biclique coverings for graphs with chromatic number k

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 Szemer\'edi who proved that the minimum is $k\log_2 k$ when $G$ is a clique.