Strong edge-colorings of sparse graphs with large maximum degree

TL;DR

This paper establishes new bounds on the strong chromatic index for sparse graphs with large maximum degree, focusing on 2-degenerate graphs and those with small maximum average degree, improving understanding of edge-coloring in such graphs.

Contribution

It provides novel upper bounds on the strong chromatic index for sparse graphs, including sharp bounds for graphs with specific degeneracy and average degree conditions.

Findings

Strong chromatic index of 2-degenerate graphs is at most 5Δ + 1.

If Mad(G) < 8/3 and Δ ≥ 9, then χs'(G) ≤ 3Δ - 3 (sharp bound).

If Mad(G) < 3 and Δ ≥ 7, then χs'(G) ≤ 3Δ.

Abstract

A {\em strong $k$-edge-coloring} of a graph $G$ is a mapping from $E(G)$ to $\{1,2,\ldots,k\}$ such that every two adjacent edges or two edges adjacent to the same edge receive distinct colors. The {\em strong chromatic index} $\chi_s'(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ admits a strong $k$-edge-coloring. We give bounds on $\chi_s'(G)$ in terms of the maximum degree $\Delta(G)$ of a graph $G$. when $G$ is sparse, namely, when $G$ is $2$-degenerate or when the maximum average degree ${\rm Mad}(G)$ is small. We prove that the strong chromatic index of each $2$-degenerate graph $G$ is at most $5\Delta(G) +1$. Furthermore, we show that for a graph $G$, if ${\rm Mad}(G)< 8/3$ and $\Delta(G)\geq 9$, then $\chi_s'(G)\leq 3\Delta(G) -3$ (the bound $3\Delta(G) -3$ is sharp) and if ${\rm Mad}(G)<3$ and $\Delta(G)\geq 7$, then $\chi_s'(G)\leq 3\Delta(G)$ (the restriction ${\rm Mad}(G)<3$ is sharp).