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Equitable colorings of Cartesian products of graphs. (English) Zbl 1241.05035

Summary: The present paper studies the following variation of vertex coloring on graphs. A graph \(G\) is equitably \(k\)-colorable if there is a mapping \(f: V(G)\to\{1,2,\dots,k\}\) such that \(f(x)\not\in f(y)\) for \(xy\in E(G)\) and \(\| f^{-1}(i)|-|f^{-1}(j)\|\leq 1\) for \(1\leq i,\,j\leq k\). The equitable chromatic number of a graph \(G\), denoted by \(\chi_=(G)\), is the minimum \(k\) such that \(G\) is equitably \(k\)-colorable. The equitable chromatic threshold of a graph \(G\), denoted by \(\chi^*_=(G)\), is the minimum \(t\) such that \(G\) is equitably \(k\)-colorable for all \(k\geq t\).
Our focus is on the equitable colorability of Cartesian products of graphs. In particular, we give exact values or upper bounds of \(\chi_= (G\square H)\) and \(\chi^*_=(G\square H)\) when \(G\) and \(H\) are cycles, paths, stars, or complete bipartite graphs.

MSC:

05C15 Coloring of graphs and hypergraphs
05C76 Graph operations (line graphs, products, etc.)
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[1] Baker, B.; Coffman, E., Mutual exclusion scheduling, Theoret. Comput. Sci., 162, 2, 225-243 (1996) · Zbl 0877.68007
[2] Blum, D.; Torrey, D.; Hammack, R., Equitable chromatic number of complete multipartite graphs, Missouri J. Math. Sci., 15, 2, 75-81 (2003) · Zbl 1027.05038
[3] Chang, G. J., A note on equitable colorings of forests, European J. Combin., 30, 809-812 (2009) · Zbl 1220.05037
[4] Chen, B.-L.; Lih, K.-W., Equitable coloring of trees, J. Combin. Theory Ser. B, 61, 1, 83-87 (1994) · Zbl 0805.05027
[5] Chen, B.-L.; Lih, K.-W.; Wu, P.-L., Equitable coloring and the maximum degree, European J. Combin., 15, 5, 443-447 (1994) · Zbl 0809.05050
[6] Chen, B.-L.; Lih, K.-W.; Yan, J.-H., Equitable coloring of interval graphs and products of graphs
[7] Erdős, P., Problem 9, (Fielder, M., Theory of Graphs and its Applications, vol. 159 (1964), Czech. Acad. Sci. Publ.: Czech. Acad. Sci. Publ. Prague)
[8] Furmańczyk, H., Equitable colorings of graph products, Opuscula Math., 26, 1, 31-44 (2006) · Zbl 1142.05025
[9] Hajnal, A.; Szemerédi, E., Proof of a conjecture of P. Erdős, (Erdős, P.; Rényi, A.; Sós, V. T., Combinatorial Theory and Applications (1970), North-Holland: North-Holland London), 601-623 · Zbl 0217.02601
[10] S. Hedetniemi, Homomorphism of graphs and automata, Univ. Michigan Technical Report 03105-44-T, 1966.; S. Hedetniemi, Homomorphism of graphs and automata, Univ. Michigan Technical Report 03105-44-T, 1966.
[11] Imrich, W.; Klavzaˇr, S., Product Graphs: Structure and Recognition (2000), Wiley: Wiley New York · Zbl 0963.05002
[12] Irani, S.; Leung, V., Scheduling with conflicts and applications to traffic signal control, (Proc. of Seventh Annu. ACM-SIAM Symp. on Discrete Algorithms (1996), SIAM: SIAM Atlanta, GA), 85-94, Philadelphia, PA · Zbl 0845.90072
[13] Janson, S.; Ruciński, A., The infamous upper tail, Random Struct. Algorithms, 20, 3, 317-342 (2002) · Zbl 0996.60023
[14] Kierstead, H. A.; Kostochka, A. V., A short proof of the Hajnal-Szemerédi theorem on equitable coloring, Combin. Probab. Comput., 17, 2, 265-270 (2008) · Zbl 1163.05015
[15] Kierstead, H. A.; Kostochka, A. V., An Ore-type theorem on equitable coloring, J. Combin. Theory Ser. B, 98, 226-234 (2008) · Zbl 1127.05039
[16] Kitagawa, F.; Ikeda, H., An existential problem of a weight-controlled subset and its application to schedule timetable construction, Discrete Math., 72, 1-3, 195-211 (1988) · Zbl 0664.90076
[17] Kostochka, A. V., Equitable colorings of outerplanar graphs, Discrete Math., 258, 1-3, 373-377 (2002) · Zbl 1009.05059
[18] Kostochka, A. V.; Nakprasit, K., Equitable coloring of \(k\)-degenerate graphs, Combin. Probab. Comput., 12, 53-60 (2003) · Zbl 1012.05063
[19] Kostochka, A. V.; Nakprasit, K., On equitable \(\Delta \)-coloring of graphs with low average degree, Theoret. Comput. Sci., 349, 1, 82-91 (2005) · Zbl 1086.05030
[20] Kostochka, A. V.; Nakprasit, K.; Pemmaraju, S. V., On equitable coloring of \(d\)-degenerate graphs, SIAM J. Discrete Math., 19, 1, 83-95 (2005) · Zbl 1082.05037
[21] Kostochka, A. V.; Pelsmajer, M. J.; West, D. B., A list analogue of equitable coloring, J. Graph Theory, 44, 3, 166-177 (2003) · Zbl 1031.05050
[22] Lam, P. C.B.; Shiu, W. C.; Tong, C. S.; Zhang, C. F., On the equitable chromatic number of complete \(n\)-partite graphs, Discrete Appl. Math., 113, 2-3, 307-310 (2001) · Zbl 0990.05050
[23] Lih, K.-W., The equitable coloring of graphs, (Du, D.-Z.; Pardalos, P., Handbook of Combinatorial Optimization, vol. 3 (1998), Kluwer: Kluwer Dordrecht), 543-566 · Zbl 0944.05049
[24] Lih, K.-W.; Wu, P.-L., On equitable coloring of bipartite graphs, Discrete Math., 151, 1-3, 155-160 (1996) · Zbl 0856.05040
[25] Lin, W.-H.; Chang, G. J., Equitable colorings of Kronecker products of graphs, Discrete Appl. Math., 158, 1816-1826 (2010) · Zbl 1208.05029
[26] Meyer, W., Equitable coloring, Amer. Math. Monthly, 80, 920-922 (1973) · Zbl 0279.05106
[27] M. Mydlarz, E. Szemerédi, Algorithmic Brooks’ theorem, Manuscript.; M. Mydlarz, E. Szemerédi, Algorithmic Brooks’ theorem, Manuscript.
[28] Pelsmajer, M. J., Equitable list-coloring for graphs of maximum degree 3, J. Graph Theory, 47, 1, 1-8 (2004) · Zbl 1053.05051
[29] S.V. Pemmaraju, Equitable colorings extend Chernoff-Hoeffding bounds, in: Proc. Fifth Internat. Workshop on Randomization and Approximation Techniques in Computer Sciences, APPROX-RANDOM 2001, pp. 285-296.; S.V. Pemmaraju, Equitable colorings extend Chernoff-Hoeffding bounds, in: Proc. Fifth Internat. Workshop on Randomization and Approximation Techniques in Computer Sciences, APPROX-RANDOM 2001, pp. 285-296. · Zbl 0998.68231
[30] Sabidussi, G., Graphs with given group and given graph-theoretical properties, Canad. J. Math., 9, 515-525 (1957) · Zbl 0079.39202
[31] Smith, B. F.; Bjorstad, P. E.; Gropp, W. D., Domain decomposition, (Parallel Multilevel Methods for Elliptic Partial Differential Equations (1996), Cambridge University Press: Cambridge University Press Cambridge), 224 · Zbl 0857.65126
[32] Tucker, A., Perfect graphs and an application to optimizing municipal services, SIAM Rev., 15, 585-590 (1973) · Zbl 0255.05111
[33] Vizing, V. G., The Cartesian product of graphs, Vyčhisl. Sistemy, 9, 30-43 (1963) · Zbl 0931.05033
[34] Wang, W.-F.; Lih, K.-W., Equitable list coloring of graphs, Taiwanese J. Math., 8, 4, 747-759 (2004) · Zbl 1063.05051
[35] Yap, H.-P.; Zhang, Y., The \(\Delta \)-equitable colouring conjecture holds for outerplanar graphs, Bull. Inst. Math. Acad. Sin., 25, 2, 143-149 (1997) · Zbl 0882.05054
[36] Yap, H.-P.; Zhang, Y., Equitable colourings of planar graphs, J. Combin. Math. Combin. Comput., 27, 97-105 (1998) · Zbl 0927.05033
[37] Zhu, X., A survey on Hedetniemi’s conjecture, Taiwanese J. Math., 2, 1, 1-24 (1998) · Zbl 0906.05024
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