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On the Tractability of (ki)-Coloring

  • Saurabh Joshi
  • Subrahmanyam KalyanasundaramEmail author
  • Anjeneya Swami Kare
  • Sriram Bhyravarapu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10743)

Abstract

In an undirected graph, a proper (ki)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (ki)-coloring problem is to compute the minimum number of colors required for a proper (ki)-coloring. This is a generalization of the classic graph coloring problem. Majumdar et al. [CALDAM 2017] studied this problem and showed that the decision version of the (ki)-coloring problem is fixed parameter tractable (FPT) with tree-width as the parameter. They asked if there exists an FPT algorithm with the size of the feedback vertex set (FVS) as the parameter without using tree-width machinery. We answer this in positive by giving a parameterized algorithm with the size of the FVS as the parameter. We also give a faster and simpler exact algorithm for \((k, k-1)\)-coloring, and make progress on the NP-completeness of specific cases of (ki)-coloring.

Notes

Acknowledgment

The authors would like to thank the anonymous reviewer for helpful comments, and pointing out a flaw in the proof of Theorem 12 in an earlier version of the paper.

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Saurabh Joshi
    • 1
  • Subrahmanyam Kalyanasundaram
    • 1
    Email author
  • Anjeneya Swami Kare
    • 1
  • Sriram Bhyravarapu
    • 1
  1. 1.Department of Computer Science and EngineeringIIT HyderabadSangareddyIndia

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