Harmonious coloring: Parameterized algorithms and upper bounds

Sudeshna Kolay, Ragukumar Pandurangan, Fahad Panolan, Venkatesh Raman, Prafullkumar Tale

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 2nnO(1) algorithm for finding a minimum harmonious coloring in split graphs improving on the naive 2O(n log n) algorithm.

Original languageEnglish
Title of host publicationGraph-Theoretic Concepts in Computer Science - 42nd International Workshop, WG 2016, Revised Selected Papers
EditorsPinar Heggernes
PublisherSpringer Verlag
Number of pages12
ISBN (Print)9783662535356
StatePublished - 1 Jan 2016
Externally publishedYes
Event42nd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2016 - Istanbul, Turkey
Duration: 22 Jun 201624 Jun 2016

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9941 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference42nd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2016

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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