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The Number of Out-Pancyclic Vertices in a Strong Tournament

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Abstract

An arc in a tournament T with n ≥ 3 vertices is called pancyclic, if it belongs to a cycle of length l for all 3 ≤ l ≤ n. We call a vertex u of T an out-pancyclic vertex of T, if each out-arc of u is pancyclic in T. Yao et al. (Discrete Appl. Math. 99, 245–249, 2000) proved that every strong tournament contains an out-pancyclic vertex. For strong tournaments with minimum out-degree 1, Yao et al. found an infinite class of strong tournaments, each of which contains exactly one out-pancyclic vertex. In this paper, we prove that every strong tournament with minimum out-degree at least 2 contains three out-pancyclic vertices. Our result is best possible since there is an infinite family of strong tournaments with minimum degree at least 2 and no more than 3 out-pancyclic vertices.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China

    Qiaoping Guo & Hongwei Li

  2. Institute of Mathematics and Applied Mathematics, Shanxi University, Taiyuan, 030006, China

    Shengjia Li & Huiling Zhao

Authors
  1. Qiaoping Guo
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  2. Shengjia Li
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  3. Hongwei Li
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  4. Huiling Zhao
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Correspondence to Qiaoping Guo.

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Guo, Q., Li, S., Li, H. et al. The Number of Out-Pancyclic Vertices in a Strong Tournament. Graphs and Combinatorics 30, 1163–1173 (2014). https://doi.org/10.1007/s00373-013-1328-6

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  • DOI: https://doi.org/10.1007/s00373-013-1328-6

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