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A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring

  • H. A. KIERSTEAD (a1) and A. V. KOSTOCHKA (a2)
Abstract

A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemerédi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r+1 colours. The proof yields a polynomial time algorithm for such colourings.

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[1]Blazewicz, J., Ecker, K., Pesch, E., Schmidt, G. and Weglarz, J. (2001) Scheduling Computer and Manufacturing Processes, 2nd edn, Springer.
[2]Chen, B.-L., Lih, K.-W. and Wu, P.-L. (1994) Equitable coloring and the maximum degree. Europ. J. Combin. 15 443447.
[3]Erdős, P. (1964) Problem 9. In Theory of Graphs and its Applications (Fiedler, M., ed.), Czech. Academy of Sciences, Prague, p. 159.
[4]Fan, G. and Kierstead, H. A. (1996) Hamiltonian square paths. J. Combin. Theory Ser. B 67 167182.
[5]Hajnal, A. and Szemerédi, E. (1970) Proof of a conjecture of P. Erdős. In Combinatorial Theory and its Application (Erdős, P., Rényi, A., and Sós, V. T., eds), North-Holland, London, pp. 601623.
[6]Janson, S. and Ruciński, A. (2002) The infamous upper tail. Random Struct. Alg. 20 317342.
[7]Komlós, J., Sárkőzy, G. and Szemerédi, E. (1998) Proof of the Seymour conjecture for large graphs. Ann. Combin. 1 4360.
[8]Kostochka, A. V., Pelsmajer, M. J. and West, D. B. (2003) A list analogue of equitable coloring. J. Graph Theory 44 166177.
[9]Kostochka, A. V. and Yu, G. (2006) Extremal problems on packing of graphs. Oberwolfach Reports 1 5557.
[10]Kostochka, A. V. and Yu, G. (2007) Ore-type graph packing problems. Combin. Probab. Comput. 16 167169.
[11]Mydlarz, M. and Szemerédi, E. Algorithmic Brooks' theorem. Manuscript.
[12]Ore, O. (1960) Note on Hamilton circuits. Amer. Math. Monthly 67 55.
[13]Pemmaraju, S. V. (2001) Equitable colorings extend Chernoff–Hoeffding bounds. In Proc. 5th International Workshop on Randomization and Approximation Techniques in Computer Science (APPROX-RANDOM 2001), pp. 285–296.
[14]Seymour, P. (1974) Problem section. In Combinatorics: Proc. British Combinatorial Conference 1973 (McDonough, T. P. and Mavron, V. C., eds), Cambridge University Press, pp. 201202.
[15]Smith, B. F., Bjorstad, P. E. and Gropp, W. D. (1996) Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press.
[16]Tucker, A. (1973) Perfect graphs and an application to optimizing municipal services. SIAM Review 15 585590.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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