Linear time algorithms for happy vertex coloring problems for trees

Abstract

Given an undirected graph G = (V,E) with |V | = n and a vertex coloring, a vertex v is happy if υ and all its neighbors have the same color.An edge is happy if its end vertices have the same color.Given a partial coloring of the vertices of the graph using k colors, the Maximum Happy Vertices (also called k-MHV) problem asks to color the remaining vertices such that the number of happy vertices is maximized.The Maximum Happy Edges (also called k-MHE) problem asks to color the remaining vertices such that the number of happy edges is maximized.For arbitrary graphs, k-MHV and k-MHE are NP-Hard for k ≥ 3.In this paper we study these problems for trees.For a fixed k we present linear time algorithms for both the problems.In general, for any k the proposed algorithms take O(nk log k) and O(nk) time respectively.