Algorithm to d-Distance Equitable Coloring of Trees

Abstract

The d-distance coloring of G (V, E) is an assignment of {1, 2, …, k} colors to the vertices such that any two vertices within distance d receive different colors. If the set of vertices of a graph G can be partitioned into k classes V₁, V₂, V₃ …, V_k, such that  for any, 1 ≤ i, j ≤ k, then G is said to be equitably k-colorable. In this paper, we introduce d-distance equitable coloring and obtain d-distance equitable chromatic number for cycle. Also, we present linear time an algorithm for d-distance coloring of trees and (|V|²) time an algorithm for d-distance equitable coloring.