Packing Chromatic Number of Distance Graphs

TL;DR

This paper investigates the packing chromatic number of infinite distance graphs with specific distance sets, providing new bounds and improving previous results for graphs with distances {1, t}.

Contribution

It offers improved bounds on the packing chromatic number for graphs with distances {1, t}, especially for large odd and even t, and refines earlier findings.

Findings

hi_{ ho}(G(Z, D))  35 for large odd t

hi_{ ho}(G(Z, D))  56 for large even t

Lower bound of 12 for t  9

Abstract

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that vertices of $G$ can be partitioned into disjoint classes $X_1, ..., X_k$ where vertices in $X_i$ have pairwise distance greater than $i$. We study the packing chromatic number of infinite distance graphs $G(Z, D)$, i.e. graphs with the set $Z$ of integers as vertex set and in which two distinct vertices $i, j \in Z$ are adjacent if and only if $|i - j| \in D$. In this paper we focus on distance graphs with $D = \{1, t\}$. We improve some results of Togni who initiated the study. It is shown that $\chi_{\rho}(G(Z, D)) \leq 35$ for sufficiently large odd $t$ and $\chi_{\rho}(G(Z, D)) \leq 56$ for sufficiently large even $t$. We also give a lower bound 12 for $t \geq 9$ and tighten several gaps for $\chi_{\rho}(G(Z, D))$ with small $t$.