The k-distance chromatic number of trees and cycles

Abstract

For any positive integer k, a k-distance coloring of a graph G is a vertex coloring of G in which no two vertices at distance less than or equal to k receive the same color. The k-distance chromatic number of G, denoted by ?<inf>k</inf>G is the smallest integer ? for which G has a k-distance ?-coloring. In this paper, we improve the lower bound for the k-distance chromatic number of an arbitrary graph for k odd case and see that trees achieve this lower bound by determining the k-distance chromatic number of trees. Also, we find k-distance chromatic number of cycles and 2-distance chromatic number of a graph G in which every pair of cycles are edge disjoint. © 2017 Kalasalingam University