Distance dominator packing coloring of type II

Abstract

In 2021, we introduced one type of the generalization of dominator coloring via packing coloring and distance domination. In this paper, we present a second type of such generalization, namely distance dominator packing coloring of type II, defined as follows. A coloring c is a k-distance dominator packing coloring of type II of G if it is a k-packing coloring of G and for each u ∈ V (G) there exists i ∈ {1, 2, 3, . . . , k} such that u c(u)-distance dominates each vertex from the color class of color i (i.e., the distance between u and all vertices from color class of color i is at most c(u)). The smallest integer k such that there exists a k-distance dominator packing coloring of G is the distance dominator packing chromatic number of type II of G, denoted by . In this paper, we provide some lower and upper bounds on the distance dominator packing chromatic number of type II, characterize connected graphs G with , and consider the relation between the packing coloring, distance dominator packing coloring of type I (introduced by Ferme and Mesarič Štesl in 2021) and distance dominator packing coloring of type II for a given graph.