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arXiv · 2205.14680

The strong chromatic index of 1-planar graphs

Yiqiao Wang·Ning Song·Jianfeng Wang·Weifan Wang
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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 $\Delta\ge 56$.

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Yiqiao Wang, Ning Song, Jianfeng Wang, Weifan Wang. 2022-05-29. The strong chromatic index of 1-planar graphs. https://doi.org/10.46298/dmtcs.9631

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