Harmonious coloring: Parameterized algorithms and upper bounds

A harmonious coloring of a graph is a partitioning of its vertex set into parts such that, there are no edges inside each part, and there is at most one edge between any pair of parts. It is known that finding a minimum harmonious coloring number is NP-hard even in special classes of graphs like trees and split graphs. We initiate a study of parameterized and exact exponential time complexity of harmonious coloring. We consider various parameterizations like by solution size, by above or below known guaranteed bounds and by the vertex cover number of the graph. While the problem has a simple quadratic kernel when parameterized by the solution size, our main result is that the problem is fixed-parameter tractable when parameterized by the size of a vertex cover of the graph. This is shown by reducing the problem to multiple instances of fixed variable integer linear programming. We also observe that it is W[1]-hard to determine whether at most n−k or Δ+1+k colors are sufficient in a harmonious coloring of an n-vertex graph G, where Δ is the maximum degree of G and k is the parameter. Concerning exact exponential time algorithms, we develop a 2 ⁿ n ^O(1) algorithm for finding a minimum harmonious coloring in split graphs improving on the naive 2 ^O(nlog⁡n) algorithm.