On clique coverings of complete multipartite graphs

A clique covering of a graph G is a set of cliques of G such that any edge of G is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number scc(G) of a graph G, is defined as the smallest possible weight of a clique covering of G. Let K-t(d) denote the complete t-partite graph with each part of size d. We prove that for any fixed d >= 2, we have lim(t ->infinity) scc(K-t(d)) = d/2t log t. This disproves a conjecture of Davoodi et al. (2016). (C) 2019 Elsevier B.V. All rights reserved.