arXiv 1508.030522015-08math.CO

大围长可平面图的强色指数的精确值

On the precise value of the strong chromatic-index of a planar graph with a large girth

Gerard Jennhwa Chang, Guan-Huei Duh

备注：: 16 pages, 2 figures

摘要：

图$G$的强$k$ -边染色是从$E(G)$到$\{1,2,\ldots,k\}$的映射，使得距离至多为2的每对不同的边接收不同的颜色。图$G$的强色指数$χ'_s(G)$是使$G$具有强$k$ -边染色的最小$k$.表示为$σ(G)=\max_{xy\in E(G)}\{\operatorname{deg}(x)+\operatorname{deg}(y)-1\}$。很容易看出，对于任何图$G$，都有$σ(G) \le χ'_s(G)$，并且当$G$是树时等式成立。对于最大度为$Δ$的可平面图$G$，利用四色定理证明了$χ'_s(G) \le 4 Δ+4$.然后在不同的$Δ$和周长条件下，将上限简化为$4Δ$，$3Δ+5$，$3Δ+1$，$3Δ$，$2Δ-1$。本文证明了：如果可平面图$G$的围长足够大，且是$σ(G)\geq Δ(G)+2$，则$G$的强色指数恰是$σ(G)$ .这个结果反映了一种直觉，即一个局部围长较大的可平面图看起来像一棵树。

英文摘要：

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 $χ'_s(G)$ of a graph $G$ is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. Denote $σ(G)=\max_{xy\in E(G)}\{\operatorname{deg}(x)+\operatorname{deg}(y)-1\}$. It is easy to see that $σ(G) \le χ'_s(G)$ for any graph $G$, and the equality holds when $G$ is a tree. For a planar graph $G$ of maximum degree $Δ$, it was proved that $χ'_s(G) \le 4 Δ+4$ by using the Four Color Theorem. The upper bound was then reduced to $4Δ$, $3Δ+5$, $3Δ+1$, $3Δ$, $2Δ-1$ under different conditions for $Δ$ and the girth. In this paper, we prove that if the girth of a planar graph $G$ is large enough and $σ(G)\geq Δ(G)+2$, then the strong chromatic index of $G$ is precisely $σ(G)$. This result reflects the intuition that a planar graph with a large girth locally looks like a tree.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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