Packing chromatic number of distance graphs

Abstract

The packing chromatic number χ ρ (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 ∈ Z are adjacent if and only if |i − j| ∈ 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 χ ρ (G(Z, D)) ≤ 35 for sufficiently large odd t and χ ρ (G(Z, D)) ≤ 56 for sufficiently large even t. We also give a lower bound 12 for t ≥ 9 and tighten several gaps for χ ρ (G(Z, D)) with small t.

Modeling the packing coloring problem of graphs

“…Lemma 1 [14]. For every packing k-coloring of a graph G and any positive integer ' satisfying 1 6 ' 6 k, it holds that X k i¼1 dðiÞ P dð1; 2; .…”

“…It follows that dð1; 2; 3; 4; 5; 6; 7; 8Þ 6 89 101 . It was established in [14] that dðiÞ ¼ 1 6iÀ9 for i > 2 since there is no pair of vertices in D 6iÀ9 ðk; tÞ with distance greater than i and hence at most one vertex of D 6iÀ9 ðk; tÞ can be colored by color i.…”

“…By using the proposed approaches, we first improve the upper bound of v q ðDð1; 4ÞÞ from 15 (see [14]) to 14. Proof.…”

“…The study of a packing coloring of distance graphs Dðk; tÞ was initiated by Togni [13] who determined the packing chromatic number of Dð1; tÞ for small values of t. Distance graphs were further investigated by Ekstein et al in [14,15], where some more results on the packing chromatic number of Dð1; tÞ and Dðk; tÞ are given.…”

“…Packing chromatic numbers and bounds for CmÃCn.For the other families of graph we apply the following concept introduced in[14]. Let D p ðk; tÞ denote a subgraph of Dðk; tÞ induced by the vertices 1; 2; .…”