Odd covers of graphs

Abstract

Given a finite simple graph (Figure presented.), an odd cover of (Figure presented.) is a collection of complete bipartite graphs, or bicliques, in which each edge of (Figure presented.) appears in an odd number of bicliques, and each nonedge of (Figure presented.) appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of (Figure presented.) by (Figure presented.) and prove that (Figure presented.) is bounded below by half of the rank over (Figure presented.) of the adjacency matrix of (Figure presented.). We show that this lower bound is tight in the case when (Figure presented.) is a bipartite graph and almost tight when (Figure presented.) 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 (Figure presented.). Babai and Frankl proposed the “odd cover problem,” which in our language is equivalent to determining (Figure presented.). In this paper, we determine that (Figure presented.) is (Figure presented.) when (Figure presented.) and is (Figure presented.) when (Figure presented.) is equivalent to 1 or (Figure presented.) modulo 8.