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The Number of Seymour Vertices in Random Tournaments and Digraphs

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Abstract

Seymour’s distance two conjecture states that in any digraph there exists a vertex (a “Seymour vertex”) that has at least as many neighbors at distance two as it does at distance one. We explore the validity of probabilistic statements along lines suggested by Seymour’s conjecture, proving that almost surely there are a “large” number of Seymour vertices in random tournaments and “even more” in general random digraphs.

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Notes

  1. This work was started by the first three authors, reported on at [9], and completed this year by Godbole and Zhang.

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Acknowledgments

Y.Z. was supported by Grant No. 14-12 from the Acheson J. Duncan Fund for the Advancement of Research in Statistics at The Johns Hopkins University. Z.C. and L.W.H. were supported by NSF Grant 0139286. A.G. was supported by NSF Grants 0139286 and 1263009. We thank the two referees for their insightful comments, and for suggesting one open question, namely “what can be said about the number of vertices that satisfies Chen et al.’s condition [3]?”

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Authors and Affiliations

  1. University of Chicago, Chicago, USA

    Zachary Cohn

  2. East Tennessee State University, Johnson City, USA

    Anant Godbole

  3. Tulane University, New Orleans, USA

    Elizabeth Wright Harkness

  4. The Johns Hopkins University, Baltimore, USA

    Yiguang Zhang

Authors
  1. Zachary Cohn
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  2. Anant Godbole
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  3. Elizabeth Wright Harkness
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  4. Yiguang Zhang
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Correspondence to Anant Godbole.

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Cohn, Z., Godbole, A., Harkness, E.W. et al. The Number of Seymour Vertices in Random Tournaments and Digraphs. Graphs and Combinatorics 32, 1805–1816 (2016). https://doi.org/10.1007/s00373-015-1672-9

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  • DOI: https://doi.org/10.1007/s00373-015-1672-9

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