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距离图的Packing染色

On Packing Colorings of Distance Graphs

Olivier Togni

发表信息：
Discrete Applied Mathematics 167 (2014) 280-289
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摘要：

图$G$的{ \em填充色数} $χ_ρ(G)$是存在从$V(G)$到$\{1,2,\ldots ,k\}$的映射$f$使得颜色$i$的任意两个顶点的距离至少为$i+1$的最小整数$k$。研究了无限距离图$G(\mathbb{Z},D)$的填充色数，即以整数集合$\mathbb{Z}$为顶点集的图，两个不同顶点$i,j\in \mathbb{Z}$相邻当且仅当$|i-j|\in D$ .我们目前的上下界$χ_ρ(G(\mathbb{Z},D))$，显示有限的$D$，包装色数是有限的。我们的主要结果涉及距离图与$D=\{1,t\}$，我们证明了一些上界的包装色数，较小的是$t\geq 447$：$χ_ρ(G(\mathbb{Z},\{1,t\}))\leq 40$，如果$t$是奇数和$χ_ρ(G(\mathbb{Z},\{1,t\}))\leq 81$，如果$t$是偶数。

英文摘要：

The {\em packing chromatic number} $χ_ρ(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This paper studies the packing chromatic number of infinite distance graphs $G(\mathbb{Z},D)$, i.e. graphs with the set $\mathbb{Z}$ of integers as vertex set, with two distinct vertices $i,j\in \mathbb{Z}$ being adjacent if and only if $|i-j|\in D$. We present lower and upper bounds for $χ_ρ(G(\mathbb{Z},D))$, showing that for finite $D$, the packing chromatic number is finite. Our main result concerns distance graphs with $D=\{1,t\}$ for which we prove some upper bounds on their packing chromatic numbers, the smaller ones being for $t\geq 447$: $χ_ρ(G(\mathbb{Z},\{1,t\}))\leq 40$ if $t$ is odd and $χ_ρ(G(\mathbb{Z},\{1,t\}))\leq 81$ if $t$ is even.

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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