On Packing Colorings of Distance Graphs

TL;DR

This paper investigates the packing chromatic number of infinite distance graphs on integers, providing bounds and specific results for graphs with distances {1,t}, showing finiteness and giving upper bounds for large t.

Contribution

It establishes bounds for the packing chromatic number of infinite distance graphs, especially for graphs with distances {1,t}, including new upper bounds for large t.

Findings

Packing chromatic number is finite for graphs with finite distance sets.

Upper bounds of 40 for odd t and 81 for even t when t ≥ 447.

Main results include bounds for graphs with distances {1,t}.

Abstract

The {\em packing chromatic number} $\chi_{\rho}(G)$ 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 $\chi_{\rho}(G(\mathbb{Z},D))$, 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 $t\geq 447$: $\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 40$ if $t$ is odd and $\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 81$ if $t$ is even.