On the Strong Chromatic Index of Sparse Graphs

TL;DR

This paper establishes new upper bounds for the strong chromatic index and list chromatic index of certain sparse planar graphs with high girth, using advanced combinatorial methods and computational techniques.

Contribution

It provides the first bounds for the strong list chromatic index of subcubic planar graphs with girth at least 41, and improves existing bounds for other classes of sparse planar graphs.

Findings

For subcubic planar graphs with girth ≥ 41, the strong list chromatic index is at most 5.

For subcubic planar graphs with girth ≥ 30, the strong chromatic index is at most 5.

For planar graphs with maximum degree 4 and girth ≥ 28, the strong chromatic index is at most 7.

Abstract

The strong chromatic index of a graph $G$, denoted $\chi_s'(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $\chi_{s,\ell}'(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors. We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with $\operatorname{girth}(G) \geq 41$ then $\chi_{s,\ell}'(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and $\operatorname{girth}(G) \geq 30$, then $\chi_s'(G) \leq 5$, improving a bound from the same paper. Finally, if $G$ is a planar graph with maximum degree at most four and $\operatorname{girth}(G) \geq 28$, then $\chi_s'(G) \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its application on the strong edge coloring, arXiv:1412.8358] in this case.