Hankel Tournaments and Special Oriented Graphs

Book Title

Discrete Mathematics and Applications

Authors

Richard Brualdi, University of Wisconsin - Madison
Lei Cao, Nova Southeastern UniversityFollow

ORCID ID

0000-0001-7613-7191

ResearcherID

G-7341-2019

Document Type

Book Chapter

ISBN

978-3-030-55857-4

Publication Date

2020

Editors

Andrei M. Raigorodskii and Michael Th. Rassias

Keywords

Tournament, Score vector, Locally transitive, Hankel tournament, Oriented graph, Cycle 2-Tournament

Description

[Chapter 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.

DOI

10.1007/978-3-030-55857-4

Publisher

Springer

First Page

109

Last Page

152

Additional Information

Mathematics Subject Classifications

05C07 05C20 05C38 05C50 15B05

Disciplines

Mathematics | Physical Sciences and Mathematics

NSUWorks Citation

Brualdi, Richard and Lei Cao. (2020). Hankel Tournaments and Special Oriented Graphs. In Andrei M. Raigorodskii and Michael Th. Rassias (Eds.), Discrete Mathematics and Applications .

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