S-packing chromatic vertex-critical graphs

Abstract For a non-decreasing sequence of positive integers S = ( s 1 , s 2 , … ) , the S -packing chromatic number χ S ( G ) of G is the smallest integer k such that the vertex set of G can be partitioned into sets X i , i ∈ [ k ] , where vertices in X i are pairwise at distance greater than s i . In this paper we introduce S -packing chromatic vertex-critical graphs, χ S -critical for short, as the graphs in which χ S ( G − u ) χ S ( G ) for every u ∈ V ( G ) . This extends the earlier concept of the packing chromatic vertex-critical graphs. We show that if G is χ S -critical, then the set { χ S ( G ) − χ S ( G − u ) : u ∈ V ( G ) } can be almost arbitrary. If G is χ S -critical and χ S ( G ) = k ( k ∈ N ), then G is called k - χ S -critical. We characterize 3- χ S -critical graphs and partially characterize 4- χ S -critical graphs when s 1 > 1 . We also deal with k - χ S -criticality of trees and caterpillars.