A note on clique immersion of strong product graphs

Abstract

Let $G,H$ be graphs, and $G⁎H$ represent a specific graph product of G and H. Define $im\left(G\right)$ as the largest t for which G contains a $K_t$-immersion. Collins, Heenehan, and McDonald posed the question: given $im\left(G\right)=t$ and $im\left(H\right)=r$, how large can $im(G⁎H)$ be? They conjectured $im(G⁎H)\ge tr$ when ⁎ denotes the strong product. In this note, we affirm that the conjecture holds for graphs with certain immersions, in particular when H contains $K_r$ as a subgraph. As a consequence we also get an alternative argument for a result of Guyer and McDonald, showing that the line graphs of constant-multiplicity multigraphs satisfy the conjecture originally proposed by Abu-Khzam and Langston.