On packing colorings of distance graphs

Abstract

The packing chromatic numberχρ(G) of a graph G is the least integer k for which there exists a mapping f from V(G) to {1,2,…,k} such that any two vertices of color i are at a distance of at least i+1. This paper studies the packing chromatic number of infinite distance graphs G(Z,D), i.e. graphs with the set Z of integers as vertex set, with two distinct vertices i,j∈Z being adjacent if and only if ∣i−j∣∈D. We present lower and upper bounds for χρ(G(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≥447: χρ(G(Z,{1,t}))≤40 if t is odd and χρ(G(Z,{1,t}))≤81 if t is even.