Bipartite Coverings and the Chromatic Number

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: • If the bipartite graphs form a partition of the edges of G, then their number is at least 2 √ log2 . This is the first improvement of the easy lower bound of log2 k, while the Alon-Saks-Seymour conjecture states that this can be improved to k − 1. • The sum of the orders of the bipartite graphs in the cover is at least (1−o(1))k log2 k. This generalizes, in asymptotic form, a result of Katona and Szemerédi who proved that the minimum is k log2 k when G is a clique.