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arXiv · 2307.16784

On bipartite coverings of graphs and multigraphs

Noga Alon
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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.

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Noga Alon. 2023-07-31. On bipartite coverings of graphs and multigraphs. https://arxiv.org/abs/2307.16784

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