Counterexamples to Clique Immersion Conjecture for Direct Products

Abstract

Let \(G\) and \(H\) be graphs, and let \(G\times H\) denote their direct product. For a graph \(G\), let \(\operatorname{im}(G)\) be the largest integer \(t\) such that \(G\) contains a \(K_t\)-immersion. Collins, Heenehan, and McDonald conjectured that if \(\operatorname{im}(G)=t\) and \(\operatorname{im}(H)=r\), then \[\operatorname{im}(G\times H)\ge (t-1)(r-1)+1.\] We disprove this conjecture by constructing an infinite family of connected bipartite counterexamples.