  A bipartite covering of a (multi)graph $G$ is a collection of bipartite
graphs, so that each edge of $G$ belongs to at least one of them. The capacity
of the covering is the sum of the numbers of vertices of these bipartite
graphs. In this note we establish a (modest) strengthening of old results of
Hansel and of Katona and Szemer\'edi, by showing that the capacity of any
bipartite covering of a graph on $n$ vertices in which the maximum size of an
independent set containing vertex number $i$ is $\alpha_i$, is at least $\sum_i
\log_2 (n/\alpha_i).$ We also obtain slightly improved bounds for a recent
result of Kim and Lee about the minimum possible capacity of a bipartite
covering of complete multigraphs.

Combinatorics;Discrete Mathematics