Distance Pattern Distinguishing Coloring of Graphs

Given a connected (p, q)− graph G = (V, E) of diameter d, ∅M ⊆ V (G) and a nonempty set X = {0, 1, ..., d} of colors of cardinality , let fM be an assignment of subsets of X to the vertices of G such that fM(u) = {d(u, v) : v ∈ M} where, d(u, v) is the usual distance between u and v . We call fM an M− distance pattern coloring of G if no two adjacent vertices have same fM. Define f ⊕ M of an edge e ∈ E(G) as  f ⊕ M(e) = fM(u) ⊕ fM(v); e = uv. A distance pattern distinguishing coloring of a graph G is an M distance pattern coloring of G such that both fM(G) and f ⊕ M(G) are injective. This paper is a study on distance pattern coloring and distance pattern distinguishing coloring of graphs.