On distance dominator packing coloring in graphs

Let G be a graph and let S = (s1,s2,..., sk) be a non-decreasing sequence of positive integers. An S-packing coloring of G is a mapping c : V(G) ? {1, 2,..., k} with the following property: if c(u) = c(v) = i, then d(u,v) > si for any i ? {1, 2,...,k}. In particular, if S = (1, 2, 3, ..., k), then S-packing coloring of G is well known under the name packing coloring. Next, let r be a positive integer and u,v ? V(G). A vertex u r-distance dominates a vertex v if dG(u, v)? r. In this paper, we present a new concept of a coloring, namely distance dominator packing coloring, defined as follows. A coloring c is a distance dominator packing coloring of G if it is a packing coloring of G and for each x ? V(G) there exists i ? {1,2, 3,...} such that x i-distance dominates each vertex from the color class of color i. The smallest integer k such that there exists a distance dominator packing coloring of G using k colors, is the distance dominator packing chromatic number of G, denoted by ?d?(G). In this paper, we provide some lower and upper bounds on the distance dominator packing chromatic number, characterize graphs G with ?d?(G) ? {2,3}, and provide the exact values of ?d?(G) when G is a complete graph, a star, a wheel, a cycle or a path. In addition, we consider the relation between ?? (G) and ?d?(G) for a graph G.