On bipartite coverings of graphs and multigraphs

07/31/2023

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by   Noga Alon, et al.

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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 Szemerédi, 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 α_i, is at least ∑_i log_2 (n/α_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.