The exponential growth of the packing chromatic number of iterated Mycielskians

Abstract

For a graph G, the packing chromatic number of G, denoted by χρ(G), is the smallest integer k such that there exists a coloring f:V(G)↦{1,…,k} of G where every two distinct vertices u and v such that f(u)=f(v) are at pairwise distance at least f(u)+1. In this paper, we study graphs G for which χρ(μt(G))=2tχρ(G) for all t≥1, where μt(G) stands for the t-iterated Mycielskian of G. We show that a first natural upper bound of χρ(μt(G)) is 2t(|V(G)|−α(G)+1) for any graph G where α(G) is the independence number of G. This bound is rounded to 2tχρ(G) if the diameter of G is two. If moreover the graph G belongs to P, class of graphs whose vertex set can be partitioned into X∪Y such that Y is an independent set, |X|<|Y| and there is an |X|-matching M such that M=xy:x∈X,y∈Y, then χρ(μt(G))=2tχρ(G). Also, we propose a special construction of a family of maximal triangle-free graph (Tn)n≥5 with typical structure and show that χρ(μt(Tn))=2tχρ(Tn).