Distance dominator packing coloring of type II

AbstractIn 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.Mathematics Subject Classification (2020): 05C1505C7005C1205C69Key words: Coloringdominationpacking coloringdominator coloringdistance dominator packing coloring