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arXiv · 0903.3048

Biclique Coverings and the Chromatic Number

Dhruv Mubayi·Sundar Vishwanathan
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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.

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Dhruv Mubayi, Sundar Vishwanathan. 2009-03-17. Biclique Coverings and the Chromatic Number. https://arxiv.org/abs/0903.3048

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