Abstract
The well-known extremal problem on hypergraph colorings is studied. We investigate whether it is possible to color a hypergraph with a fixed number of colors equitably, i.e., so that, on the one hand, no edge is monochromatic and, on the other hand, the cardinalities of the color classes are almost the same. It is proved that if H = (V,E) is a simple hypergraph in which the least cardinality of an edge equals k, |V| = n, r|n, and
where c > 0 is an absolute constant, then there exists an equitable r-coloring of H.
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References
P. Erdős and A. Hajnal, “On a property of families of sets,” ActaMath. Acad. Sci. Hungar 12, 87–123 (1961).
P. Erdős, “On a combinatorial problem,” NordiskMatematisk Tidsskrift 11, 5–10 (1963).
P. Erdős, “On a combinatorial problem. II,” ActaMath. Acad. Sci. Hungar 15, 445–447 (1964).
A. V. Kostochka, “Color-critical graphs and hypergraphs with few edges: a survey,” in Bolyai Soc. Math. Stud., Vol. 15: More Sets, Graphs and Numbers (Springer-Verlag, Berlin, 2006), pp. 175–198.
A. M. Raigorodskii and D. A. Shabanov, “The Erdős–Hajnal problem of hypergraph colouring, its generalizations, and related problems,” UspekhiMat. Nauk 66 5, 109–182 (2011) [RussianMath. Surveys 66 5, 933–1002 (2011)].
P. Erdős and L. Lovász, “Problems and results on 3-chromatic hypergraphs and some related questions,” in Colloq. Math. Soc. Janos Bolyai, Vol. 10: Infinite and Finite Sets (North Holland, Amsterdam, 1975), Vol. II, pp. 609–627.
J. Beck, “On 3-chromatic hypergraphs,” DiscreteMath. 24 2, 127–137 (1978).
L. Lu, On a problem of Erdős and Lovász on coloring non-uniform hypergraphs, wwwmathscedu/~lu/papers/ propertyB. pdf.
D. A. Shabanov, “Around Erdős–Lovász problemon colorings of non-uniform hypergraphs,” DiscreteMath. 338 11, 1976–1981 (2015).
D. A. Shabanov, “Coloring non-uniform hypergraphs without short cycles,” Graphs Combin. 30 5, 1249–1260 (2014).
A. Hajnal and E. Szemerédi, “Proof of a conjecture of P. Erdős,” in Combinatorial Theory and Its Applications (North-Holland, Amsterdam, 1970), pp. 601–623.
L. Lu and L. Székely, “Using Lovász Local Lemma in the space of random injections,” Electron. J. Combin. 14 (Research Paper R63) (2007).
D. A. Shabanov, “Equitable two-colorings of uniform hypergraphs,” European J. Combin. 43, 185–203 (2015).
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Original Russian Text © I. R. Shirgazina, 2016, published in Matematicheskie Zametki, 2016, Vol. 99, No. 3, pp. 441–456.
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Shirgazina, I.R. Equitable colorings of nonuniform hypergraphs. Math Notes 99, 444–456 (2016). https://doi.org/10.1134/S0001434616030147
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DOI: https://doi.org/10.1134/S0001434616030147