论文标题
$ s $ - 包装距离图的颜色$ g（\ mathbb {z}，\ {2，t \}）$
$S$-packing colorings of distance graphs $G(\mathbb{Z},\{2,t\})$
论文作者
论文摘要
给定图形$ g $和非删除序列$ s =（a_1，a_2，\ ldots）$的正整数，地图$ f：v（g）\ rightarrow \ {1，\ ldots，k \ \} $是$ s $ s $ k $ g $ g $ g $ g $ g $ g） $ f（u）= f（v）= i $ $ u $和$ v $ in $ g $的距离大于$ a_i $。最小的$ k $使得$ g $具有$ s $包装$ k $ - 颜色是$ s $包装的色度，$χ_s（g）$，$ g $。在本文中，我们考虑距离图$ g（\ mathbb {z}，\ {2，t \}）$，其中$ t> 1 $是一个奇数的整数，它具有$ \ mathbb {z} $作为其顶点set，$ i，$ i，$ i，j \ in \ mathbb {z} $ at j {我们确定图形的$ s $包装色数$ g（\ mathbb {z}，\ {2，t \}）$，其中$ s $是所有$ i $的$ a_i \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in $ i $的任何序列。此外，我们给出了$ d $ d $ distance色的距离图$ g（\ mathbb {z}，\ {2，t \}）$的下限和上限，在$ d \ ge t-3 $中，它给出了确切的值。还讨论了对相应的$ s $包装的循环图的含义。
Given a graph $G$ and a non-decreasing sequence $S=(a_1,a_2,\ldots)$ of positive integers, the mapping $f:V(G) \rightarrow \{1,\ldots,k\}$ is an $S$-packing $k$-coloring of $G$ if for any distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=i$ the distance between $u$ and $v$ in $G$ is greater than $a_i$. The smallest $k$ such that $G$ has an $S$-packing $k$-coloring is the $S$-packing chromatic number, $χ_S(G)$, of $G$. In this paper, we consider the distance graphs $G(\mathbb{Z},\{2,t\})$, where $t>1$ is an odd integer, which has $\mathbb{Z}$ as its vertex set, and $i,j\in\mathbb{Z}$ are adjacent if $|i-j|\in\{2,t\}$. We determine the $S$-packing chromatic numbers of the graphs $G(\mathbb{Z},\{2,t\})$, where $S$ is any sequence with $a_i\in\{1,2\}$ for all $i$. In addition, we give lower and upper bounds for the $d$-distance chromatic numbers of the distance graphs $G(\mathbb{Z},\{2,t\})$, which in the cases $d\ge t-3$ give the exact values. Implications for the corresponding $S$-packing chromatic numbers of the circulant graphs are also discussed.
