List strong edge-coloring of graphs with maximum degree 4

TL;DR

This paper proves that for graphs with maximum degree 4, the list strong chromatic index is at most 22, extending previous bounds and confirming a conjecture for this case.

Contribution

It establishes the upper bound of 22 for the list strong chromatic index of graphs with maximum degree 4, extending Cranston's result to list colorings.

Findings

The upper bound for list strong chromatic index when Δ=4 is 22.

The result confirms the conjecture for Δ=4.

Extension of previous bounds to list coloring case.

Abstract

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $\chi_{s}'(G)$ which is the minimum number of colors that allow a strong edge-coloring of $G$. Erd\H{o}s and Ne\v{s}et\v{r}il conjectured in 1985 that the upper bound of $\chi_{s}'(G)$ is $\frac{5}{4}\Delta^{2}$ when $\Delta$ is even and $\frac{1}{4}(5\Delta^{2}-2\Delta +1)$ when $\Delta$ is odd, where $\Delta$ is the maximum degree of $G$. The conjecture is proved right when $\Delta\leq3$. The best known upper bound for $\Delta=4$ is 22 due to Cranston previously. In this paper we extend the result of Cranston to list strong edge-coloring, that is to say, we prove that when $\Delta=4$ the upper bound of list strong chromatic index is 22.