$k$-intersection edge-coloring subcubic planar multigraphs

TL;DR

This paper investigates the $k$-intersection edge-coloring of subcubic planar multigraphs, establishing bounds for the 2-intersection chromatic index with multiplicity constraints, extending known results from simple graphs.

Contribution

It extends the concept of $k$-intersection chromatic index to multigraphs with bounded edge multiplicity and proves a sharp upper bound for subcubic planar multigraphs with multiplicity two.

Findings

The 2-intersection chromatic index of such multigraphs is at most 5.

This bound is proven to be sharp.

Extension of known simple graph results to multigraphs.

Abstract

Given an edge-coloring of a simple graph, assign to every vertex $v$ a set $S_v$ comprised of the colors used on the edges incident to $v$. The $k$-intersection chromatic index of a graph is the minimum $t$ such that the edge set can be properly $t$-colored, additionally requiring that for every two adjacent vertices $u$ and $v$, $|S_u \cap S_v| \le k$. For all $k \neq 2$, this value is known for subcubic planar graphs, and furthermore, these values are best possible. We naturally extend this definition to multigraphs with bounded edge multiplicity, and we show that every subcubic planar multigraph with edge multiplicity at most two has 2-intersection chromatic index at most 5, which is sharp.