On odd covers of cliques and disjoint unions

TL;DR

This paper investigates the minimum size of collections of bipartite graphs needed to cover edges of complete graphs and other structures with odd multiplicity, advancing understanding of odd graph covers.

Contribution

It determines the exact minimum size of odd covers for complete graphs under new conditions and explores odd covers of unions of odd cliques and cycles.

Findings

Minimum odd cover size for complete graphs when n is odd or ≡ 18 mod 24

Exact values for odd covers of unions of odd cliques and cycles

Extended the theory of odd graph covers beyond complete graphs

Abstract

Babai and Frankl posed the ``odd cover problem" of finding the minimum cardinality of a collection of complete bipartite graphs such that every edge of the complete graph of order $n$ is covered an odd number of times. In a previous paper with O'Neill, some of the authors proved that this value is always $\lceil n / 2 \rceil$ or $\lceil n / 2 \rceil + 1$ and that it is the former whenever $n$ is a multiple of $8$. In this paper, we determine this value to be $\lceil n / 2 \rceil$ whenever $n$ is odd or equivalent to $18$ modulo $24$. We also further the study of odd covers of graphs which are not complete, wherein edges are covered an odd number of times and nonedges an even number of times by the complete bipartite graphs in the collection. Among various results on disjoint unions, we find the minimum cardinality of an odd cover of a union of odd cliques and of a union of cycles.