Abstract
A k-coloring of a graph is an assignment of integers between 1 and k to vertices in the graph such that the endpoints of each edge receive different numbers. We study a local variation of the coloring problem, which imposes further requirements on every set of three vertices: We are not allowed to use two consecutive numbers for a path on three vertices, or three consecutive numbers for a cycle on three vertices. Given a graph G and a positive integer k, the local coloring problem asks for whether G admits a local k-coloring. We show that it cannot be solved in subexponential time, unless the Exponential Time Hypothesis fails, and a new reduction for the NP-hardness of this problem. Our structural observations behind these reductions are of independent interests. We close the paper with a short remark on local colorings of perfect graphs.
Supported in part by the Hong Kong Research Grants Council (RGC) under grant 152261 and the National Natural Science Foundation of China (NSFC) under grants 61572414 and 61672536.
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Notes
- 1.
Chartrand et al. [1] stated the first part as Theorem 2.4, and mentioned the second part immediately after the proof of this theorem.
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You, J., Cao, Y., Wang, J. (2019). Local Coloring: New Observations and New Reductions. In: Chen, Y., Deng, X., Lu, M. (eds) Frontiers in Algorithmics. FAW 2019. Lecture Notes in Computer Science(), vol 11458. Springer, Cham. https://doi.org/10.1007/978-3-030-18126-0_5
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DOI: https://doi.org/10.1007/978-3-030-18126-0_5
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