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arXiv · 2202.09822

Odd Covers of Graphs

Calum Buchanan·Alexander Clifton·Eric Culver·Jiaxi Nie·Jason O'Neill·Puck Rombach·Mei Yin
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Abstract

Given a finite simple graph $G$, an odd cover of $G$ is a collection of complete bipartite graphs, or bicliques, in which each edge of $G$ appears in an odd number of bicliques and each non-edge of $G$ appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of $G$ by $b_2(G)$ and prove that $b_2(G)$ is bounded below by half of the rank over $\mathbb{F}_2$ of the adjacency matrix of $G$. We show that this lower bound is tight in the case when $G$ is a bipartite graph and almost tight when $G$ is an odd cycle. However, we also present an infinite family of graphs which shows that this lower bound can be arbitrarily far away from $b_2(G)$. Babai and Frankl (1992) proposed the "odd cover problem," which in our language is equivalent to determining $b_2(K_n)$. Radhakrishnan, Sen, and Vishwanathan (2000) determined $b_2(K_n)$ for an infinite but density zero subset of positive integers $n$. In this paper, we determine $b_2(K_n)$ for a density $3/8$ subset of the positive integers.

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Calum Buchanan, Alexander Clifton, Eric Culver, Jiaxi Nie, Jason O'Neill, Puck Rombach, Mei Yin. 2022-02-20. Odd Covers of Graphs. https://arxiv.org/abs/2202.09822

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