Abstract
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 0 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.
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