About the number of directed paths in tournaments

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.

Introduction

The question of counting the exact number of Hamiltonian paths in a tournament is scarcely treated in the literature. However, when it comes to find approximations for this number, the situation is different. Several mathematicians have worked over the years on bounding the number of Hamiltonian paths in a tournament, and on characterizing tournaments with minimum or maximum numbers of Hamiltonian paths or cycles. It is easy to see that the transitive tournament has the minimum number of directed Hamiltonian paths, which is exactly one. More than seventy years ago, Szele gave upper and lower bounds for the maximum number $h_{max}\left(n\right)$ of directed Hamiltonian paths in a tournament on $n$ vertices. He proved the following:

Theorem 1 [9]

Let $n$ be an integer such that $n\ge 2$. The number $h_{max}\left(n\right)$ is bounded as follows:

$$\frac{n!}{2^{n-1}}\le h_{max}\left(n\right)\le c_1\frac{n!}{2^{\frac{3}{4}n}},$$

where $c_1$ is a positive constant independent of $n$.

In [1], Alon improved the upper bound for $h_{max}\left(n\right)$. He proved that $h_{max}\left(n\right)\le c_2\cdot n^{3∕2}\frac{n!}{2^{n-1}}$.

If we limit our study to strong tournaments, then the problem of determining the minimum number of directed Hamiltonian paths becomes a complex one. Moon and Busch worked on establishing approximations for the minimum number $h_{min}\left(n\right)$ of directed Hamiltonian paths in a strong tournament on $n$ vertices. In [5], Moon gave upper and lower bounds for $h_{min}\left(n\right)$. He proved the following:

Theorem 2 [5]

$${\alpha }^{n-1}\le h_{min}\left(n\right)\le \left\{\begin{array}{ccc}\hfill 3\cdot {\beta }^{n-3}\hfill & \hfill \text{for}\hfill & \hfill n\equiv 0(mod3),\hfill \\ \hfill {\beta }^{n-1}\hfill & \hfill \text{for}\hfill & \hfill n\equiv 1(mod3),\hfill \\ \hfill 9\cdot {\beta }^{n-5}\hfill & \hfill \text{for}\hfill & \hfill n\equiv 2(mod3),\hfill \end{array}\right.$$

where $\alpha =\sqrt[4]{6}$ and $\beta =\sqrt[3]{5}$.

Busch improved this result in [2] by giving the exact value of the minimum $h_{min}\left(n\right)$ for a strong tournament on $n$ vertices as follows:

$$h_{min}\left(n\right)=\left\{\begin{array}{ccc}\hfill 3\cdot {\beta }^{n-3}\hfill & \hfill \text{for}\hfill & \hfill n\equiv 0(mod3),\hfill \\ \hfill {\beta }^{n-1}\hfill & \hfill \text{for}\hfill & \hfill n\equiv 1(mod3),\hfill \\ \hfill 9\cdot {\beta }^{n-5}\hfill & \hfill \text{for}\hfill & \hfill n\equiv 2(mod3).\hfill \end{array}\right.$$

Later on, Moon and Yang [6] gave a characterization of the strong tournaments realizing this minimum, which they called special chains of nearly transitive tournaments. Thomassen also gave lower bounds to the number of directed Hamiltonian paths in a strong tournament, with conditions on the minimal in or out degree of the tournament. He mentioned in [10] that it is not hard to prove that if $T$ is a strong tournament with minimal out-degree at least $k$, then $T$ contains at least $k!$ Hamiltonian paths starting from each vertex of $T$. Moreover, he proved that in a strong tournament with minimal in-degree $k$, there are at least $2^{k-1}$ Hamiltonian paths starting from each vertex of this tournament.

Among other results, as mentioned in the abstract, we prove that the tournament realizing $h_{max}\left(n\right)$ should be strong. We also focus on the properties of tournaments having given numbers of directed Hamiltonian paths and we study the number of directed Hamiltonian paths under some particular conditions.