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On colorings of non-uniform hypergraphs without short cycles. (English) Zbl 1274.05174

Nešetřil, Jarik (ed.) et al., Extended abstracts of the sixth European conference on combinatorics, graph theory and applications, EuroComb 2011, Budapest, Hungary, August 29 – September 2, 2011. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 38, 749-754 (2011).
Summary: The work deals with a generalization of Erdős-Lovász problem concerning colorings of non-uniform hypergraphs. Let \(H=(V,E)\) be a hypergraph and let \(f_r(H)=\sum_{e\in E}r^{1-|e|}\) for some \(r\geq 2\). Erdős and Lovász asked about the value \(f(n)\) equal to the minimum possible value of \(f_2(H)\) where \(H\) is a 3-chromatic hypergraph with minimum edge-cardinality \(n\). In our paper we study a similar problem in the class of hypergraphs with large girth.
For the entire collection see [Zbl 1242.05003].

MSC:

05C15 Coloring of graphs and hypergraphs
05C65 Hypergraphs

References:

[1] Erdos, P., and Lovász, L., Problems and results on 3-chromatic hypergraphs and some related questions, Infinite and Finite Sets, Colloquia Mathematica Societatis Janos Bolyai 10 (1973), 609-627.
[2] Beck, J.: On 3-chromatic hypergraphs. Discrete mathematics 24, 127-137 (1978) · Zbl 0429.05055
[3] Lu, L., On a problem of Erdos and Lovász on coloring non-uniform hypergraphs, submitted, preprint is available at www.math.sc.edu/ lu/papers/.
[4] Radhakrishnan, J.; Srinivasan, A.: Improved bounds and algorithms for hypergraph two-coloring. Random structures and algorithms 16, 4-32 (2000) · Zbl 0942.05024
[5] Spencer, J. H.: Coloring n-sets red and blue. J. combinatorial theory, series A 30, 112-113 (1981) · Zbl 0448.05032
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