  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.
