Odd Covers of Graphs

TL;DR

This paper investigates the minimum number of complete bipartite subgraphs needed to cover the edges of a graph with parity conditions, establishing bounds and exact values for specific graph classes and subsets of integers.

Contribution

It introduces a new lower bound for the odd cover number based on the adjacency matrix rank and determines this number for certain classes of graphs and subsets of integers.

Findings

Lower bound for $b_2(G)$ based on adjacency matrix rank over $ extbf{F}_2$

Exact values of $b_2(G)$ for bipartite graphs and odd cycles

Determination of $b_2(K_n)$ for a density $3/8$ subset of positive integers

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.