A characterization of 4-χS-vertex-critical graphs for packing sequences with s1=1 and s2≥3

Abstract

If S=(s1,s2,…) is a non-decreasing sequence of positive integers, then the S-packing k-coloring of a graph G is a mapping c:V(G)→[k] such that if c(u)=c(v)=i for u≠v∈V(G), then dG(u,v)>si. The S-packing chromatic number of G is the smallest integer k such that G admits an S-packing k-coloring. A graph G is χS-vertex-critical if χS(G−u)<χS(G) for each u∈V(G). If G is χS-vertex-critical and χS(G)=k, then G is k−χS-vertex-critical. In this paper, 4−χS-vertex-critical graphs are characterized for sequences S=(1,s2,s3,…) with s2≥3. There are 28 sporadic examples and two infinite families of such graphs.