On Clique Coverings of Complete Multipartite Graphs

TL;DR

This paper determines the asymptotic behavior of the sigma clique cover number for complete t-partite graphs with fixed part size, disproving a previous conjecture and providing new insights into clique coverings.

Contribution

The paper establishes the exact asymptotic limit of the sigma clique cover number for complete t-partite graphs with fixed part size, challenging prior conjectures.

Findings

scc(K_t(d)) asymptotically equals (d/2) t log t as t approaches infinity

Disproves the conjecture of Davoodi, Javadi, and Omoomi

Provides new asymptotic bounds for clique coverings in multipartite graphs

Abstract

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 $d \ge 2$, we have $$\lim_{t \rightarrow \infty} scc(K_t(d))= \frac{d}{2} t\log t.$$ This disproves a conjecture of Davoodi, Javadi and Omoomi.