The strong chromatic index of 1-planar graphs
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909461449277440 |
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| author | Wang, Yiqiao Song, Ning Wang, Jianfeng Wang, Weifan |
| author_facet | Wang, Yiqiao Song, Ning Wang, Jianfeng Wang, Weifan |
| contents | 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 $Δ\ge 56$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_14680 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The strong chromatic index of 1-planar graphs Wang, Yiqiao Song, Ning Wang, Jianfeng Wang, Weifan Combinatorics 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 $Δ\ge 56$. |
| title | The strong chromatic index of 1-planar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2205.14680 |