Equitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture

Abstract

Chen, Lih, and Wu conjectured that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are K r,r (for odd r) and K r+1. If true, this would be a strengthening of the Hajnal-Szemerédi Theorem and Brooks’ Theorem. We extend their conjecture to disconnected graphs. For r ≥ 6 the conjecture says the following: If an r-colorable graph G with maximum degree r is not equitably r-colorable then r is odd, G contains K r,r and V(G) partitions into subsets V 0, …, V t such that G[V 0] = K r,r and for each 1 ≤ it, G[V i ] = K r . We characterize graphs satisfying the conclusion of our conjecture for all r and use the characterization to prove that the two conjectures are equivalent. This new conjecture may help to prove the Chen-Lih-Wu Conjecture by induction.

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References

  1. [1]

    N. Alon and Z. Füredi: Spanning subgraphs of random graphs, Graphs and Combinatorics8 (1992), 91–94.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    N. Alon and R. Yuster: Almost H-factors in dense graphs, Graphs and Combinatorics8 (1992), 95–102.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    N. Alon and R. Yuster: H-factors in dense graphs, J. Combinatorial Theory, Ser. B66 (1996), 269–282.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    J. Blazewicz, K. Ecker, E. Pesch, G. Schmidt and J. Weglarz: Scheduling computer and manufacturing processes, 2nd ed., Springer, Berlin, 485 p. (2001).

    MATH  Google Scholar 

  5. [5]

    R. L. Brooks: On coloring the nodes of a network, Proc. Cambridge Phil. Soc.37 (1941), 194–197.

    Article  MathSciNet  Google Scholar 

  6. [6]

    B.-L. Chen, K.-W. Lih and P.-L. Wu: Equitable coloring and the maximum degree, Europ. J. Combinatorics15 (1994), 443–447.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    G. Dirac: Some theorems on abstract graphs, Proc. London Math. Soc.2 (1952), 69–81.

    Article  MathSciNet  Google Scholar 

  8. [8]

    A. Hajnal and E. Szemerédi: Proof of a conjecture of P. Erdős, in: Combinatorial Theory and its Application (P. Erdős, A. Rényi and V. T. Sós, eds.), pp. 601–623, North-Holland, London, 1970.

    Google Scholar 

  9. [9]

    S. Janson and A. Ruciński: The infamous upper tail, Random Structures and Algorithms20 (2002), 317–342.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    H. A. Kierstead and A. V. Kostochka: A Short Proof of the Hajnal-Szemerédi Theorem on Equitable Coloring, Combin., Prob. and Comput.17 (2008), 265–270.

    MATH  MathSciNet  Google Scholar 

  11. [11]

    J. Komlós, G. Sárközy and E. Szemerédi: Proof of the Seymour conjecture for large graphs, Annals of Combinatorics1 (1998), 43–60.

    Article  Google Scholar 

  12. [12]

    A. V. Kostochka and K. Nakprasit: On equitable Δ-coloring of graphs with low average degree, Theor. Comp. Sci.349 (2005), 82–91.

    MATH  MathSciNet  Google Scholar 

  13. [13]

    A. V. Kostochka and G. Yu: Extremal problems on packing of graphs, Oberwolfach Reports No. 1 (2006), 55–57.

  14. [14]

    K.-W. Lih and P.-L. Wu: On equitable coloring of bipartite graphs, Discrete Math.151 (1996), 155–160.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    S. V. Pemmaraju: Equitable colorings extend Chernoff-Hoeffding bounds, in: Proceedings of the 5th International Workshop on Randomization and Approximation Techniques in Computer Science (APPROX-RANDOM 2001), pp. 285–296, 2001.

  16. [16]

    V. Rödl and A. Ruciński: Perfect matchings in ɛ-regular graphs and the blow-up lemma, Combinatorica19(3) (1999), 437–452.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    B. F. Smith, P. E. Bjøstad and W. D. Gropp: Domain decomposition; Parallel multilevel methods for elliptic partial differential equations, Cambridge University Press, Cambridge, 224 p. (1996).

    MATH  Google Scholar 

  18. [18]

    A. Tucker: Perfect graphs and an application to optimizing municipal services, SIAM Review15 (1973), 585–590.

    MATH  Article  MathSciNet  Google Scholar 

  19. [19]

    H.-P. Yap and Y. Zhang: The equitable Δ-colouring conjecture holds for outerplanar graphs, Bull. Inst. Math. Acad. Sin.5 (1997), 143–149.

    MathSciNet  Google Scholar 

  20. [20]

    H.-P. Yap and Y. Zhang: Equitable colourings of planar graphs, J. Comb. Math. Comb. Comp.27 (1998), 97–105.

    MATH  MathSciNet  Google Scholar 

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Correspondence to Henry A. Kierstead.

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Research of this author is supported in part by the NSA grant MDA 904-03-1-0007.

Research of this author is supported in part by the NSF grant DMS-0650784 and by grant 06-01-00694 of the Russian Foundation for Basic Research.

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Kierstead, H.A., Kostochka, A.V. Equitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture. Combinatorica 30, 201–216 (2010). https://doi.org/10.1007/s00493-010-2420-7

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Mathematics Subject Classification (2000)

  • 05C15
  • 05C35