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Coloring Non-uniform Hypergraphs Without Short Cycles

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Abstract

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 \({{f_r(H)=\sum\limits_{e \in E}r^{1-|e|}}}\) for some r ≥ 2. Erdős and Lovász proposed to find the value f (n) equal to the minimum possible value of f 2(H) where H is 3-chromatic hypergraph with minimum edge-cardinality n. In the paper we study similar problem for the class of hypergraphs with large girth. We prove that if H is a hypergraph with minimum edge-cardinality n ≥ 3 and girth at least 4, satisfying the inequality

$$f_r(H) \leq \frac{1}{2}\, \left(\frac{n}{{\rm ln}\, n}\right)^{2/3},$$

then H is r -colorable. Our result improves previous lower bounds for f (n) in the class of hypergraphs without 2- and 3-cycles.

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Authors and Affiliations

  1. Department of Probability Theory, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, GSP-1, Leninskie Gory, 1, 119991, Moscow, Russia

    Dmitry A. Shabanov

  2. Department of Discrete Mathematics, Faculty of Innovations and High Technology, Moscow Institute of Physics and Technology, 9, Institutskii per, Dolgoprudny, 141700, Moscow Region, Russia

    Dmitry A. Shabanov

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Correspondence to Dmitry A. Shabanov.

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Shabanov, D.A. Coloring Non-uniform Hypergraphs Without Short Cycles. Graphs and Combinatorics 30, 1249–1260 (2014). https://doi.org/10.1007/s00373-013-1333-9

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