Abstract
A clique of a graph G is a set of pairwise adjacent vertices of G. A clique-coloring of G is an assignment of colors to the vertices of G in such a way that no inclusion-wise maximal clique of size at least two of G is monochromatic. An equitable clique-coloring of G is a clique-coloring such that any two color classes differ in size by at most one. Bacsó and Tuza proved that connected claw-free graphs with maximum degree at most four, other than chordless odd cycles of order greater than three, are 2-clique-colorable and a 2-clique-coloring can be found in \(O(n^{2})\) Bacsó and Tuza (Discrete Math Theor Comput Sci 11(2):15–24, 2009). In this paper we prove that every connected claw-free graph with maximum degree at most four, not a chordless odd cycle of order greater than three, has an equitable 2-clique-coloring. In addition we improve the algorithm described in the paper mentioned by giving an equitable 2-clique-coloring in linear time for this class of graphs.
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The authors would like to thank the referees for valuable comments and suggestions.
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This research was supported by National Nature Science Foundation of China (Grant Nos. 11601262, 11801361) and Shanghai Nature Science Foundation of China(No. 18ZR1416300).
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This research was partially supported by National Nature Science Foundation of China (Nos. 11601262 and 11801361) and Shanghai Nature Science Foundation of China (No. 18ZR1416300).
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Liang, Z., Dong, Y., Zhao, Y. et al. Equitable Clique-Coloring in Claw-Free Graphs with Maximum Degree at Most 4. Graphs and Combinatorics 37, 445–454 (2021). https://doi.org/10.1007/s00373-020-02253-x
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DOI: https://doi.org/10.1007/s00373-020-02253-x