On clique immersions in line graphs

TL;DR

This paper explores how clique immersions in line graphs behave under certain graph transformations, proving new results about their properties and confirming a conjecture for simple graphs.

Contribution

It establishes a relation between clique immersions in line graphs and their edge-multiplied versions, confirming a conjecture for simple graphs and analyzing immersion-minor equivalences.

Findings

If $L(G)$ immerses $K_t$, then $L(mG)$ immerses $K_{mt}$.

For line graphs, $K_t$-immersion is equivalent to $K_t$-minor for $t \\leq 4$, but not for $t \\geq 5$.

Abstract

We prove that if $L(G)$ immerses $K_t$ then $L(mG)$ immerses $K_{mt}$, where $mG$ is the graph obtained from $G$ by replacing each edge in $G$ with a parallel edge of multiplicity $m$. This implies that when $G$ is a simple graph, $L(mG)$ satisfies a conjecture of Abu-Khzam and Langston. We also show that when $G$ is a line graph, $G$ has a $K_t$-immersion iff $G$ has a $K_t$-minor whenever $t\leq 4$, but this equivalence fails in both directions when $t \geq 5$.