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On Sullivan’s conjecture on cycles in 4-free and 5-free digraphs

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Abstract

For a simple digraph G, let β(G) be the size of the smallest subset X ⊆ E(G) such that G−X has no directed cycles, and let γ(G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This paper proves that β(G) ≤ 0.3819γ(G) if G is a 4-free digraph, and β(G) ≤ 0.2679γ(G) if G is a 5-free digraph. These improve the results of Sullivan in 2008.

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Authors and Affiliations

  1. Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, P. R. China

    Hao Liang

  2. School of Mathematical Sciences, University of Science and Technology of China, Wentsun Wu Key Laboratory of CAS, Hefei, 230026, P. R. China

    Jun Ming Xu

Authors
  1. Hao Liang
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  2. Jun Ming Xu
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Corresponding author

Correspondence to Hao Liang.

Additional information

The first author is supported by the Key Project of Chinese Ministry of Education (Grant No. 109140); the second author is supported by NNSF of China (Grant No. 11071233)

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Liang, H., Xu, J.M. On Sullivan’s conjecture on cycles in 4-free and 5-free digraphs. Acta. Math. Sin.-English Ser. 29, 53–64 (2013). https://doi.org/10.1007/s10114-012-1538-5

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  • DOI: https://doi.org/10.1007/s10114-012-1538-5

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