About the number of directed paths in tournaments
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 of directed Hamiltonian paths in a tournament on vertices. He proved the following:
Theorem 1 [9] Let be an integer such that . The number is bounded as follows: where is a positive constant independent of .
In [1], Alon improved the upper bound for . He proved that .
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 of directed Hamiltonian paths in a strong tournament on vertices. In [5], Moon gave upper and lower bounds for . He proved the following:
Theorem 2 [5] where and .
Busch improved this result in [2] by giving the exact value of the minimum for a strong tournament on vertices as follows:
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 is a strong tournament with minimal out-degree at least , then contains at least Hamiltonian paths starting from each vertex of . Moreover, he proved that in a strong tournament with minimal in-degree , there are at least Hamiltonian paths starting from each vertex of this tournament.
Among other results, as mentioned in the abstract, we prove that the tournament realizing 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.
Section snippets
Parities of directed paths starting or ending with a specified vertex
We first introduce a simple yet important lemma that we will need later. Recall that the acyclic ordering of the strong components of a tournament is the ordering of these components such that for any , all arcs are directed from towards . In this case, we may write . We denote by the number of directed Hamiltonian paths in a tournament .
Lemma 3 If is the acyclic ordering of the strong components of a tournament, then
Proof Let be a
Tournaments with given number of directed Hamiltonian paths
In this section, we assume conditions on the number of directed Hamiltonian paths in a tournament , and deduce some properties of . We first prove that the maximality of is a sufficient condition for the tournament to be strong. Next, we characterize tournaments having directed Hamiltonian paths. It is known that the minimal number of Hamiltonian paths in a tournament is , and that is transitive. Since is always odd, it would be interesting to think
Number of directed Hamiltonian paths in special tournaments
In this section, we give formulas for the number of directed Hamiltonian paths in some tournaments obtained from the nearly transitive tournament by reversing particular arcs. The basic idea that inspired this work is the simple, yet effective method, used to compute the number of directed Hamiltonian paths in the nearly transitive tournament. The same idea allows us, by considering many cases, to generalize that formula to more complicated tournaments. We have seen that if we reverse the arc
Acknowledgment
We would like to thank Laszlo Szalay for guiding us to C. Sanna’s article and for pointing out the link between -adic numbers and our work.
References (10)
On the number of Hamiltonian cycles in tournaments
Discrete Math.
(1980)The maximum number of Hamiltonian paths in tournaments
Combinatorica
(1990)A note on the number of Hamiltonian paths in strong tournaments
Electron. J. Combin.
(2006)Chemins et circuits hamiltoniens des graphes complets
C. R. Math. Acad. Sci. Paris
(1959)Catalan’s conjecture: Another old diophantine problem solved
Bull. (New Ser.) Amer. Math. Soc.
(2003)