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On the Maximum Number of Spanning Copies of an Orientation in a Tournament

Part of: Graph theory

Published online by Cambridge University Press:  06 June 2017

RAPHAEL YUSTER
RAPHAEL YUSTER*
Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel (e-mail: raphy@math.haifa.ac.il)
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Abstract

For an orientation H with n vertices, let T(H) denote the maximum possible number of labelled copies of H in an n-vertex tournament. It is easily seen that T(H) ≥ n!/2e(H), as the latter is the expected number of such copies in a random tournament. For n odd, let R(H) denote the maximum possible number of labelled copies of H in an n-vertex regular tournament. In fact, Adler, Alon and Ross proved that for H=Cn, the directed Hamilton cycle, T(Cn) ≥ (e−o(1))n!/2n, and it was observed by Alon that already R(Cn) ≥ (e−o(1))n!/2n. Similar results hold for the directed Hamilton path Pn. In other words, for the Hamilton path and cycle, the lower bound derived from the expectation argument can be improved by a constant factor. In this paper we significantly extend these results, and prove that they hold for a larger family of orientations H which includes all bounded-degree Eulerian orientations and all bounded-degree balanced orientations, as well as many others. One corollary of our method is that for any fixed k, every k-regular orientation H with n vertices satisfies T(H) ≥ (ek−o(1))n!/2e(H), and in fact, for n odd, R(H) ≥ (ek−o(1))n!/2e(H).

MSC classification

Primary: 05C20: Directed graphs (digraphs), tournaments
Secondary: 05C35: Extremal problems 05C45: Eulerian and Hamiltonian graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 5 , September 2017 , pp. 775 - 796
Copyright
Copyright © Cambridge University Press 2017 

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