Abstract
An equitable k-partition (\(k\ge 2\)) of a vertex set S is a partition of S into k subsets (may be empty sets) such that the sizes of any two subsets of S differ by at most one. A maximal-m-clique is a clique with m vertices which is not in a larger clique than itself. A local-equitable k-coloring of G is an assignment of k colors to the vertices of G such that, for every maximal clique of G, the coloring of this clique forms an equitable k-partition of itself. Local-equitable coloring of graphs is a generalization of proper vertex coloring of graphs. In \(K_{4}\)-free planar graphs, the local-equitable 3-coloring is precisely the same as the proper 3-vertex-coloring. The famous Grötzsch Theorem states that triangle-free planar graphs are 3-colorable. In this paper we show that maximal-3-clique-free planar graphs are local-equitable 3-colorable, which is a generalization of Grötzsch Theorem.
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References
G. Bacsó, Zs. Tuza, Clique-transversal sets and weak 2-colorings in graphs of small maximum degree, Disc. Math. Theoret. Comput. Sci. 11(2), 15–24 (2009)
Bacsó, G., Gravier, S., Gyárfás, A., Preissmann, M., Sebő, A.: Coloring the maximal cliques of graphs. SIAM J. Disc. Math. 17, 361–376 (2004)
G. Bacsó, Z. Ryjá\(\check{c}\)ek, Zs. Tuza, Coloring the cliques of line graphs, Discr. Math. 340, 2641–2649 (2007)
Charbit, P., Penev, I., Thomassé, S., Trotignon, N.: Perfect graphs of arbitrarily large clique-chromatic number. J. Combin. Theory Ser. B 116, 456–464 (2016)
Chudnovsky, M., Lo, I.: Decomposing and clique-coloring (diamond, odd-hole)-free graphs. J. Graph Theory 86, 5–41 (2017)
Défossez, D.: Clique coloring some classess of odd-hole-free graphs. J. Graph Theory 53, 233–249 (2006)
Défossez, D.: Complexity of clique coloring odd-hole-free graphs. J. Graph Theory 62, 139–156 (2009)
H. Grötzsch, Ein dreifarbensatz, Ein dreifarbensatz fur dreikreisfreienetze auf der kugel, Math. Nat. Reihe 8 109–120 (1959)
J. Kratochvíl, Zs. Tuza, On the complexity of bicoloring clique hypergraphs of graphs. J. Algorithms 45, 40-54 (2002)
Liang, Z., Shan, E., Kang, L.: Clique-coloring claw-free graphs. Graph Combin. 32, 1473–1488 (2016)
Liang, Z., Shan, E., Zhang, Y.: A linear-time algorithm for clique-coloring problem in circular-arc graphs. J. Comb. Optim. 33, 147–155 (2017)
Liang, Z., Wu, J., Shan, E.: List-coloring clique-hypergraphs of \(K_5\)-minor-free graphs strongly. Discrete Math. 343, 111777 (2020)
Liang, Z., Wang, J., Cai, J., Yang, X.: On the complexity of local-equitable coloring of graphs. Theoret. Comput. Sci. 906, 76–82 (2022)
Marx, D.: Complexity of clique coloring and related problems. Theor. Comput. Sci. 412, 3487–3500 (2011)
B. Mohar, R. \(\check{S}\)krekovski, The Grötzsch theorem for the hypergraph of maximal cliques, Electr. J. Combin. 6, #R26 (1999)
Penev, I.: Perfect graphs with no balanced skew-partition are 2-clique-colorable. J. Graph Theory 81, 213–235 (2016)
Shan, E., Liang, Z., Kang, L.: Clique-transversal sets and clique-coloring in planar graphs. Eur. J. Combin. 36, 367–376 (2014)
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The funding has been received from Nature Science Foundation of Shandong Province with Grant nos. ZR2021MA012 and ZR2021MA103; the Natural Science Research Foundation of Colleges and Universities of Anhui Province with Grant no. KJ2021A0968.
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This research was partially supported by Nature Science Foundation of Shandong Province (Nos. ZR2021MA012 and ZR2021MA103) and the Natural Science Research Foundation of Colleges and Universities of Anhui Province (KJ2021A0968).
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Liang, Z., Wang, J. & Bai, C. A Generalization of Grötzsch Theorem on the Local-Equitable Coloring. Graphs and Combinatorics 39, 4 (2023). https://doi.org/10.1007/s00373-022-02598-5
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DOI: https://doi.org/10.1007/s00373-022-02598-5