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- W4298085835 abstract "A strong $k$-edge-coloring of a graph $G$ is a mapping from $E(G)$ to ${1,2,ldots,k}$ such that every pair of distinct edges at distance at most two receive different colors. The strong chromatic index $chi'_s(G)$ of a graph $G$ is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. Denote $sigma(G)=max_{xyin E(G)}{operatorname{deg}(x)+operatorname{deg}(y)-1}$. It is easy to see that $sigma(G) le chi'_s(G)$ for any graph $G$, and the equality holds when $G$ is a tree. For a planar graph $G$ of maximum degree $Delta$, it was proved that $chi'_s(G) le 4 Delta +4$ by using the Four Color Theorem. The upper bound was then reduced to $4Delta$, $3Delta+5$, $3Delta+1$, $3Delta$, $2Delta-1$ under different conditions for $Delta$ and the girth. In this paper, we prove that if the girth of a planar graph $G$ is large enough and $sigma(G)geq Delta(G)+2$, then the strong chromatic index of $G$ is precisely $sigma(G)$. This result reflects the intuition that a planar graph with a large girth locally looks like a tree." @default.
- W4298085835 created "2022-10-01" @default.
- W4298085835 creator A5002508116 @default.
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- W4298085835 date "2015-08-12" @default.
- W4298085835 modified "2023-09-26" @default.
- W4298085835 title "On the precise value of the strong chromatic-index of a planar graph with a large girth" @default.
- W4298085835 doi "https://doi.org/10.48550/arxiv.1508.03052" @default.
- W4298085835 hasPublicationYear "2015" @default.
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