About the number of directed paths in tournaments

摘要

In this article, we prove some properties about the number of directed paths in tournaments. We first prove that if T is a tournament on n vertices and we choose a vertex v in T, then the total number of directed paths starting with v, of lengths between O and n - 2, is congruent, mod 2, to the number of directed paths of the same lengths, ending with v. Next, we prove that if the number of directed Hamiltonian paths in a tournament T is maximal, then T must be strong. Then, we study some properties of tournaments where the number of directed Hamiltonian paths is a power of 3. Finally, we compute the exact number of directed Hamiltonian paths in some special tournaments, obtained from the nearly transitive tournament by reversing extra arcs. (C) 2019 Elsevier B.V. All rights reserved.