On clique immersions in line graphs
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 AbuKhzam 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 $t \geq 5$.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.07964
 Bibcode:
 2019arXiv190907964G
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 11 pages, 5 figures