$S$-packing colorings of distance graphs $G(\mathbb{Z},\{2,t\})$

TL;DR

This paper determines the $S$-packing chromatic numbers for a class of infinite distance graphs with specific adjacency rules, providing exact values and bounds, and discusses implications for related circulant graphs.

Contribution

It offers exact $S$-packing chromatic numbers for $G( extbf{Z}, extbf{\{2,t\}})$ with sequences in extbf{\{1,2\}}, and establishes bounds for $d$-distance chromatic numbers, extending understanding of graph colorings.

Findings

Exact $S$-packing chromatic numbers for $G( extbf{Z}, extbf{\{2,t\}})$ with $a_i ext{ in } extbf{\{1,2\}}$.

Lower and upper bounds for $d$-distance chromatic numbers, exact for $d ext{ } extgreater= t-3$.

Implications for $S$-packing chromatic numbers of circulant graphs.

Abstract

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, $\chi_S(G)$, 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.