(untitled) (Statistics, Optimization & Information Computing, 2024)
All graphs in this paper are simple and connected. Let G=(V,E) be a graph where V(G) is nonempty of vertex set of G and E(G) is possibly empty set of unordered pairs of elements of V(G). The distance from u to v in G is the length of a shortest path joining them, denoted by d(u,v). For some positive integer k, a function c:V(G)→ {1,2,...k} is called packing k-coloring if any two not adjacent vertices u and v, c(u)=c(v)=i and d(u,v)≥ i+1. The minimum number k such that the graph G has a packing k-coloring is called the packing chromatic number, denoted by χ_ρ(G) . In this paper, we investigate the packing chromatic number of some family trees, namely centipede, firecracker, broom, double star and banana tree graphs.