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Hypergraphs with No Cycle of a Given Length


Recently, the authors gave upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge. In the present paper we extend this bound to m-uniform hypergraphs (for all m ≥ 3), as well as m-uniform hypergraphs avoiding a cycle of length 2k. Finally we consider non-uniform hypergraphs avoiding cycles of length 2k or 2k + 1. In both cases we can bound |h| by O(n1+1/k) under the assumption that all h ∈ ε() satisfy |h| ≥ 4k2.

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[1]B. Bollobás and E. Győri (2008) Pentagons vs. triangles. Discrete Math. 308 43324336.

[2]J. A. Bondy and M. Simonovits (1974) Cycles of even length in graphs. J. Combin. Theory Ser. B 16 97105.

[3]P. Erdős and T. Gallai (1959) On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hungar. 10 337356.

[4]A. Gyárfás , M. Jacobson , A. Kézdy and J. Lehel (2006) Odd cycles and Theta-cycles in hypergraphs. Discrete Math. 306 24812491.

[7]A. Kostochka and J. Verstraete (2005) Even cycles in hypergraphs. J. Combin. Theory Ser. B 94 173182.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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