Dominating sets in k-majority tournaments

  • Authors:

  • Noga Alon;Graham Brightwell;H. A. Kierstead;A. V. Kostochka;Peter Winkler

  • Affiliations:

  • Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel;Department of Mathematics, London School of Economics, London, UK;Department of Mathematics and Statistics, Arizona State University, Tempe, AZ;Department of Mathematics, University of Illinois, Urbana, IL and Institute of Mathematics, Novosibirsk, Russia;Department of Mathematics, Dartmouth College, Hanover, NH and Bell Labs, Lucent Technologies and at Institute for Advanced Study

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2006

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Abstract

A k-majority tournament T on a finite vertex set V is defined by a set of 2k - 1 linear orderings of V, with u → v if and only if u lies above v in at least k of the orders. Motivated in part by the phenomenon of "non-transitive dice", we let F(k) be the maximum over all k-majority tournaments T of the size of a minimum dominating set of T.We show that F(k) exists for all k 0, that F(2) = 3 and that in general C1k/log k ≤ F(k) ≤ C2k log k for suitable positive constants C1 and C2.