The strong chromatic index of 1-planar graphs

TL;DR

This paper establishes an upper bound on the strong chromatic index of graphs based on their maximum average degree and degree, specifically improving bounds for 1-planar graphs with high maximum degree.

Contribution

The paper introduces a new bound for the strong chromatic index of graphs using maximum average degree, and improves existing bounds for 1-planar graphs with large maximum degree.

Findings

very graph with bounded maximum average degree has a strong chromatic index bounded by a function of its maximum degree.

or 1-planar graphs with maximum degree t least 56, the strong chromatic index is at most 14 times the maximum degree.

n improved upper bound on the strong chromatic index for 1-planar graphs compared to previous results.

Abstract

The chromatic index $\chi'(G)$ of a graph $G$ is the smallest $k$ for which $G$ admits an edge $k$-coloring such that any two adjacent edges have distinct colors. The strong chromatic index $\chi'_s(G)$ of $G$ is the smallest $k$ such that $G$ has an edge $k$-coloring with the condition that any two edges at distance at most 2 receive distinct colors. A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, we show that every graph $G$ with maximum average degree $\bar{d}(G)$ has $\chi'_{s}(G)\le (2\bar{d}(G)-1)\chi'(G)$. As a corollary, we prove that every 1-planar graph $G$ with maximum degree $\Delta$ has $\chi'_{\rm s}(G)\le 14\Delta$, which improves a result, due to Bensmail et al., which says that $\chi'_{\rm s}(G)\le 24\Delta$ if $\Delta\ge 56$.