The -distance chromatic number of trees and cycles

For any positive integer , a -distance coloring of a graph is a vertex coloring of in which no two vertices at distance less than or equal to receive the same color. The -distance chromatic number of , denoted by is the smallest integer for which has a -distance -coloring. In this paper, we improve the lower bound for the -distance chromatic number of an arbitrary graph for odd case and see that trees achieve this lower bound by determining the -distance chromatic number of trees. Also, we find -distance chromatic number of cycles and 2-distance chromatic number of a graph in which every pair of cycles are edge disjoint.