Open Distance Pattern Coloring of a Graph

Given a connected -graph of diameter and a nonempty set of colors of cardinality let be an assignment of subsets of to the vertices of such that where is the usual distance between and Given such a function for all vertices in an induced edge function of an edge We call an -open distance pattern coloring of if no two adjacent vertices have same and if such an exists for a graph then is called an open distance pattern colorable graph; the minimum cardinality of such an if it exists, is the open distance pattern coloring number of denoted by Also, if for all we call a -uniform -open distance pattern colorable graph and if such an exists for a graph then is called -uniform open distance pattern colorable graph. In this paper we initiate a study on properties of graphs which belongs to this coloring class.