On Odd Covers of Cliques and Disjoint Unions

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 is covered an odd number of times. In a previous paper with O'Neill, some of the authors proved that this value is always or and that it is the former whenever is a multiple of 8. In this paper, we determine this value to be whenever 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.