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 v 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 \(k \ge 3\). 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.