d2-coloring of a Graph

A subset S of V is called an i-set(i  2) if no two vertices in S have the distance i. The 2-set number  (G) of a graph is the maximum cardinality among all 2-sets of G. A d 2 -coloring of a graph is an assignment of colors to its vertices so that no two vertices have the distance two get the same color. The d 2 2 -chromatic number  (G) of a graph G is the minimum number of d 2 d 2 -colors need to G. In this paper, we initiate a study of these two new parameters.