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.

翻译：一个（多重）图$G$的二部覆盖是一组二部图的集合，使得$G$的每条边至少属于其中一个二部图。该覆盖的容量是这些二部图的顶点数之和。本文对Hansel以及Katona与Szemerédi的经典结果进行了（适度的）加强：若一个$n$阶图中包含顶点编号$i$的最大独立集大小为$\alpha_i$，则其任意二部覆盖的容量至少为$\sum_i \log_2 (n/\alpha_i)$。对于Kim与Lee关于完全多重图二部覆盖最小容量的近期结果，我们也获得了略优的界值。