Packing chromatic vertex-critical graphs

Abstract

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ∈ [k], where vertices in Vi are pairwise at distance at least i + 1. Packing chromatic vertex-critical graphs, χρ-critical for short, are introduced as the graphs G for which χρ(G − x) < χρ(G) holds for every vertex} can be almost arbitrary. The 3-χρ-critical graphs are characterized, and 4-χρcritical graphs are characterized in the case when they contain a cycle of length at least 5 which is not congruent to 0 modulo 4. It is shown that for every integer k ≥ 2 there exists a k-χρ-critical tree and that a k-χρ-critical caterpillar exists if and only if k ≤ 7. Cartesian products are also considered and in particular it is proved that if G and H are vertex-transitive graphs and diam(G) + diam(H) ≤ χρ(G), then G H is χρ-critical.