# On the Tractability of (k, i)-Coloring

• Saurabh Joshi
• Subrahmanyam Kalyanasundaram
• 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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## 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