Abstract
Let the vertices of a graph G be colored with k colors such that no adjacent vertices receive the same color and the sizes of the color classes differ by at most one. Then G is said to be equitably k-colorable. The equitable chromatic number x = (G) is the smallest integer k such that G is equitably k-colorable. In this article, we survey recent progress on the equitable coloring of graphs. We pay more attention to work done on the Equitable ∆-Coloring Conjecture. We also discuss related graph coloring notions and their problems. The survey ends with suggestions for further research topics.
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References
Bollobás and R. K. Guy, Equitable and proportional coloring of trees, J. Combin. Theory Ser. B, 34 (1983) 177–186.
R. L. Brooks, On colouring the nodes of a network, Proc. Cambridge Philos. Soc., 37 (1941) 194–197.
B.-L. Chen, M.-T. Ko, and K.-W. Lih, Equitable and m-bounded coloring of split graphs, in M. Deza, R. Euler, and I. Manoussakis, eds., Combinatorics and Computer Science, Lecture Notes in Computer Science 1120, (Springer, Berlin, 1996 ) 1–5.
B.-L. Chen and K.-W. Lih, A note on the m-bounded chromatic number of a tree, Europ. J. Combinatorics, 14 (1993) 311–312.
B.-L. Chen and K.-W. Lih, Equitable coloring of trees, J. Combin. Theory Ser B, 61 (1994) 83–87.
B.-L. Chen, K.-W. Lih, and P.-L. Wu, Equitable coloring and the maximum degree, Europ. J. Combinatorics, 15 (1994) 443–447.
B.-L. Chen, K.-W. Lih, and J.-H. Yan, Equitable coloring of graph products, manuscript, 1998.
B.-L. Chen, K.-W. Lih, and J.-H. Yan, A note on equitable coloring of interval graphs, manuscript, 1998.
B.-L. Chen and C.-H. Wu, The equitable coloring of complete partite graphs, manuscript, 1994.
H.-L. Fu, Some results on equalized total coloring, Congressus Numerantium, 102 (1994) 111–119.
R. P. Gupta, On decompositions of a multigraph into spanning sub-graphs, Bull. Amer. Math. Soc., 80 (1974) 500–502.
R. K. Guy, Monthly research problems, 1969–1975, Amer. Math. Monthly, 82 (1975) 995–1004.
A. Hajnal and E. Szemerédi, Proof of a conjecture of Erdös, in: A. Rényi and V. T. Sos, eds., Combinatorial Theory and Its Applications, Vol. II, Colloq. Math. Soc. Janos Bolyai 4 ( North-Holland, Amsterdam, 1970 ) 601–623.
P. Hansen, A. Hertz, and J. Kuplinsky, Bounded vertex colorings of graphs, Discrete Math., 111 (1993) 305–312.
A. J. W. Hilton and D. de Werra, A sufficient condition for equitable edge-colourings of simple graphs, Discrete Math., 128 (1994) 179–201.
T. R. Jensen and B. Toft, Graph Coloring Problems, ( John Wiley & Sons, New York, 1995 ).
K.-W. Lih and P.-L. Wu, On equitable coloring of bipartite graphs, Discrete Math., 151 (1996) 155–160.
C.-Y. Lin, The Equitable Coloring of Bipartite Graphs, Master thesis, Tunghai University, Taiwan, 1995.
A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, ( Academic Press, New York, 1979 ).
W. Meyer, Equitable coloring, Amer. Math. Monthly, 80 (1973) 920–922.
B. Toft, 75 graph-colouring problems, in: R. Nelson and R. J. Wilson, eds., Graph Colourings, ( Longman, Essex, 1990 ) 9–36.
A. C. Tucker, Perfect graphs and an application to optimizing municipal services, SIAM Rev., 15 (1973) 585–590.
V. G. Vizing, On an estimate of the chromatic class of a p-graph (Russian), Diskret. Analiz., 3 (1964) 25–30.
W. Wang, Equitable Colorings and Total Colorings of Graphs, Ph.D. dissertation, Nanjing University, China, 1997.
D. de Werra, Some uses of hypergraph in timetabling, Asis-Pacific J. Oper. Res., 2 (1985) 2–12.
H. P. Yap and Y. Zhang, On equitable coloring of graphs, manuscript, 1996.
H. P. Yap and Y. Zhang, The equitable 0-colouring conjecture holds for outerplanar graphs, Bull. Inst. Math. Acad. Sinica, 25 (1997) 143–149.
Y.Zhang and H. P. Yap, equitable colorings of planar graphs, to appear in J. Combin. Math. Combin. Comput.
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Lih, KW. (1998). The Equitable Coloring of Graphs. In: Du, DZ., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0303-9_31
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DOI: https://doi.org/10.1007/978-1-4613-0303-9_31
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