Abstract
In an undirected graph, a proper (k, i)-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 (k, i)-coloring problem is to compute the minimum number of colors required for a proper (k, i)-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 (k, i)-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 (k, i)-coloring.
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Even though [12] claims a running time of \(O((^q_k)^{tw} n^{O(1)})\) for their algorithm, there is an additional factor of \(\left( {\begin{array}{c}q\\ k\end{array}}\right) \) that is omitted, presumably because \(\left( {\begin{array}{c}q\\ k\end{array}}\right) \) is treated as a constant.
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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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Joshi, S., Kalyanasundaram, S., Kare, A.S., Bhyravarapu, S. (2018). On the Tractability of (k, i)-Coloring. In: Panda, B., Goswami, P. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2018. Lecture Notes in Computer Science(), vol 10743. Springer, Cham. https://doi.org/10.1007/978-3-319-74180-2_16
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DOI: https://doi.org/10.1007/978-3-319-74180-2_16
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