Odd Covers of Graphs

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 b2(G) and prove that b2(G) is bounded below by half of the rank over F2 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 b2(G).Babai and Frankl (1992) proposed the "odd cover problem," which in our language is equivalent to determining b2(Kn). Radhakrishnan, Sen, and Vishwanathan (2000) determined b2(Kn) for an infinite but density zero subset of positive integers n. In this paper, we determine b2(Kn) for a density 3/8 subset of the positive integers.

Shortest-Path Kernels on Graphs

“…This problem can be generalized to odd-covers of graphs by any family F of graphs instead of paths and cycles. Clique odd-covers were studied in [11], while biclique odd-covers were studied in [10]. The proof techniques in [10,11] include minimum rank arguments, vertex covers, linear algebra, and forbidden subgraphs.…”

“…Clique odd-covers were studied in [11], while biclique odd-covers were studied in [10]. The proof techniques in [10,11] include minimum rank arguments, vertex covers, linear algebra, and forbidden subgraphs. They are quite different from the techniques used in this paper, highlighting that different approaches can be successful for odd-cover problems depending on the family of covering graphs considered.…”

“…This question was motivated as the binary field adaptation of the celebrated linear algebraic proof of Graham and Pollak [22,23] that one needs at least n − 1 edge-disjoint complete bipartite graphs to decompose K n . Babai and Frankl's question was generalized in [10] to finding the minimum cardinality of a biclique odd-cover of an arbitrary graph G. Relatedly, one can replace complete bipartite graphs by complete graphs to obtain the minimum cardinality of a clique odd-cover of G. Clique odd-covers are studied under the name subgraph complementation systems in [11]; these were motivated by a question posed by Vatter [25] on representing a graph G as a sum of cliques modulo 2, and they are equivalent to finding orthogonal representations of G over F 2 [26].…”