The strong chromatic index of 1-planar graphs

TL;DR

This paper studies the strong edge coloring problem by connecting the strong chromatic index $χ'_{s}(G)$ to the maximum average degree $\bar{d}(G)$. The authors prove the general bound $χ'_{s}(G) ≤ (2\bar{d}(G)-1)χ'(G)$ through edge-color class contraction and degeneracy arguments, and show that for simple graphs this bound yields practical colorings. As corollaries, they obtain $χ'_{s}(G) ≤ 14Δ$ for every 1-planar graph of maximum degree $Δ$, and refined bounds for special 1-planar families: IC-planar graphs satisfy $χ'_{s}(G) ≤ 6Δ+20$ and optimal 1-planar graphs satisfy $χ'_{s}(G) ≤ 10Δ+14$. The results illuminate how degenerate structures and average degree govern strong edge colorings and open questions about the optimal constants in these bounds.

Abstract

The chromatic index $χ'(G)$ 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 $χ'_s(G)$ 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 $χ'_{s}(G)\le (2\bar{d}(G)-1)χ'(G)$. As a corollary, we prove that every 1-planar graph $G$ with maximum degree $Δ$ has $χ'_{\rm s}(G)\le 14Δ$, which improves a result, due to Bensmail et al., which says that $χ'_{\rm s}(G)\le 24Δ$ if $Δ\ge 56$.

Theorems & Definitions (32)

• Conjecture 1
• Lemma 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• proof