The Packing Coloring of Distance Graphs $D(k,t)$

TL;DR

This paper investigates the packing chromatic number of infinite distance graphs D(k,t), providing bounds and generalizations of previous results, especially for large t and various parity conditions of k and t.

Contribution

It extends prior work on D(1,t) to more general D(k,t) graphs, establishing new upper bounds and bounds for small parameters.

Findings

Proved that for large t, the packing chromatic number is at most 30 when both k and t are odd.

Established an upper bound of 56 for cases where exactly one of k, t is odd.

Provided bounds for the packing chromatic number for small values of k and t.

Abstract

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $p$ such that vertices of $G$ can be partitioned into disjoint classes $X_{1}, ..., X_{p}$ where vertices in $X_{i}$ have pairwise distance greater than $i$. For $k < t$ we study the packing chromatic number of infinite distance graphs $D(k, t)$, 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 $|i - j| \in \{k, t\}$. We generalize results by Ekstein et al. for graphs $D (1, t)$. For sufficiently large $t$ we prove that $\chi_{\rho}(D(k, t)) \leq 30$ for both $k$, $t$ odd, and that $\chi_{\rho}(D(k, t)) \leq 56$ for exactly one of $k$, $t$ odd. We also give some upper and lower bounds for $\chi_{\rho}(D(k, t))$ with small $k$ and $t$. Keywords: distance graph; packing coloring; packing chromatic number