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 ≤ i ≤ t, 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.
References
N. Alon and Z. Füredi: Spanning subgraphs of random graphs, Graphs and Combinatorics8 (1992), 91–94.
N. Alon and R. Yuster: Almost H-factors in dense graphs, Graphs and Combinatorics8 (1992), 95–102.
N. Alon and R. Yuster: H-factors in dense graphs, J. Combinatorial Theory, Ser. B66 (1996), 269–282.
J. Blazewicz, K. Ecker, E. Pesch, G. Schmidt and J. Weglarz: Scheduling computer and manufacturing processes, 2nd ed., Springer, Berlin, 485 p. (2001).
R. L. Brooks: On coloring the nodes of a network, Proc. Cambridge Phil. Soc.37 (1941), 194–197.
B.-L. Chen, K.-W. Lih and P.-L. Wu: Equitable coloring and the maximum degree, Europ. J. Combinatorics15 (1994), 443–447.
G. Dirac: Some theorems on abstract graphs, Proc. London Math. Soc.2 (1952), 69–81.
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.
S. Janson and A. Ruciński: The infamous upper tail, Random Structures and Algorithms20 (2002), 317–342.
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.
J. Komlós, G. Sárközy and E. Szemerédi: Proof of the Seymour conjecture for large graphs, Annals of Combinatorics1 (1998), 43–60.
A. V. Kostochka and K. Nakprasit: On equitable Δ-coloring of graphs with low average degree, Theor. Comp. Sci.349 (2005), 82–91.
A. V. Kostochka and G. Yu: Extremal problems on packing of graphs, Oberwolfach Reports No. 1 (2006), 55–57.
K.-W. Lih and P.-L. Wu: On equitable coloring of bipartite graphs, Discrete Math.151 (1996), 155–160.
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.
V. Rödl and A. Ruciński: Perfect matchings in ɛ-regular graphs and the blow-up lemma, Combinatorica19(3) (1999), 437–452.
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).
A. Tucker: Perfect graphs and an application to optimizing municipal services, SIAM Review15 (1973), 585–590.
H.-P. Yap and Y. Zhang: The equitable Δ-colouring conjecture holds for outerplanar graphs, Bull. Inst. Math. Acad. Sin.5 (1997), 143–149.
H.-P. Yap and Y. Zhang: Equitable colourings of planar graphs, J. Comb. Math. Comb. Comp.27 (1998), 97–105.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-010-2420-7