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

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!/2ⁿ, and it was observed by Alon that already R(C n) ≥ (e-o(1))n!/2ⁿ. 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).