On bipartite coverings of graphs and multigraphs

TL;DR

This paper improves bounds on the capacity of bipartite coverings of graphs and multigraphs, relating it to independent set sizes and providing tighter bounds for specific cases.

Contribution

It strengthens existing bounds on bipartite covering capacity, linking it to vertex-specific independent set sizes and refining results for complete multigraphs.

Findings

Lower bound on bipartite covering capacity based on independent set sizes.

Improved bounds for bipartite coverings of complete multigraphs.

Extension of classical results by Hansel, Katona, and Szemerédi.

Abstract

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.