## 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!/2^{e(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=C n, the directed Hamilton cycle, T(C n) ≥ (e-o(1))n!/2^{n}, and it was observed by Alon that already R(C n) ≥ (e-o(1))n!/2^{n}. Similar results hold for the directed Hamilton path P n. 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) ≥ (e ^{k}-o(1))n!/2^{e(H)}, and in fact, for n odd, R(H) ≥ (e ^{k}-o(1))n!/2^{e(H)}.

Original language | American English |
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Pages (from-to) | 775-796 |

Number of pages | 22 |

Journal | Combinatorics Probability and Computing |

Volume | 26 |

Issue number | 5 |

DOIs | |

State | Published - 1 Sep 2017 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics