Counterexamples to a 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.