Hankel Tournaments and Special Oriented Graphs

Richard A. Brualdi () and Lei Cao ()
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Richard A. Brualdi: University of Wisconsin
Lei Cao: Halmos College of Arts and Natural Sciences, Nova Southeastern University

A chapter in Discrete Mathematics and Applications, 2020, pp 109-152 from Springer

Abstract: Abstract A Hankel tournament T of order n (an n × n Hankel tournament matrix T = [t ij]) is a tournament such that i → j an edge implies (n + 1 − j) → (n + 1 − i) is also an edge (t ij = t n+1−j,n+1−i) for all i and j. Hankel tournament matrices are (0, 1)-matrices which are combinatorially antisymmetric about the main diagonal and symmetric about the Hankel diagonal (the antidiagonal). Locally transitive tournaments are tournaments such that the in-neighborhood and the out-neighborhood of each vertex are transitive. Tournaments form a special class of oriented graphs. The score vectors of Hankel tournaments and of locally transitive tournaments have been characterized where each score vector of a locally transitive tournament is also a score vector of a Hankel tournament. In this paper we continue investigations into Hankel tournaments and locally transitive tournaments. We investigate Hankel cycles in Hankel tournaments and show in particular that a strongly connected Hankel tournament contains a Hankel Hamilton cycle and, in fact, is Hankel “even-pancyclic” or Hankel “odd-pancyclic.” We show that a Hankel score vector can be achieved by a Hankel “half-transitive” tournament, extending the corresponding result for score vectors of tournaments. We also consider some results on oriented graphs and the question of attainability of prescribed degrees by oriented graphs. Finally, we extend some results on 2-tournaments to Hankel 2-tournaments. In some instances we rely on the reader to extend arguments already in the literature. We illustrate our investigations with many examples.

Keywords: Tournament; Score vector; Locally transitive; Hankel tournament; Oriented graph; Cycle; 2-Tournament; 05C07; 05C20; 05C38; 05C50; 15B05 (search for similar items in EconPapers)
Date: 2020
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-55857-4_5

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DOI: 10.1007/978-3-030-55857-4_5

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