On distance dominator packing coloring in graphs

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