Hamiltonian Paths In Tournaments Research Articles

A short proof of the bijection between minimal feedback arc sets and Hamiltonian paths in tournaments

A short proof of the bijection between minimal feedback arc sets and Hamiltonian paths in tournaments

About the number of directed paths in tournaments

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.

Finding Hamiltonian paths in tournaments on clusters

This paper presents a general methodology for the communication-efficient parallelization of graph algorithms using the divide-and-conquer approach and shows that this class of problems can be solved in cluster environments with good communication efficiency. Specifically, the first practical parallel algorithm, based on a general coarse-grained model, for finding Hamiltonian paths in tournaments is presented. On any such parallel machines, this algorithm uses only (3log p+1), where p is the number of processors, communication rounds, which is independent of the tournament size, and can reuse the existing linear-time algorithm in the sequential setting. For theoretical completeness, the algorithm is revised for fine-grained models, where the ratio of computation and communication throughputs is low or the local memory size, $$O(\frac{N}{p})$$ , of each individual processor is extremely limited $$(\frac{N}{p} \ge p^\epsilon,$$ for any $$\epsilon > 0)$$ , solving the problem with O(log p) communication rounds, while the hidden constant grows with the scalability factor 1/?. Experiments have been carried out on a Linux cluster of 32 Sun Ultra5 computers and an SGI Origin 2000 with 32 R10000 processors. The algorithm performance on the Linux Cluster reaches 75% of the performance on the SGI Origin 2000 when the tournament size is about one million.

On the maximum number of Hamiltonian paths in tournaments

Solving an old conjecture of Szele we show that the maximum number of directed Hamiltonian paths in a tournament onn vertices is at mostc · n3/2· n!/2n−1, wherec is a positive constant independent ofn.

On the maximum number of Hamiltonian paths in tournaments

AbstractBy using the probabilistic method, we show that the maximum number of directed Hamiltonian paths in a complete directed graph with n vertices is at least (e−o(1)) (n!/2n−1). © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 291–296, 2001

Oriented Hamiltonian Paths in Tournaments: A Proof of Rosenfeld's Conjecture

We prove that with three exceptions, every tournament of order n contains each oriented path of order n. The exceptions are the antidirected paths in the 3-cycle, in the regular tournament on 5 vertices, and in the Paley tournament on 7 vertices.

The maximum number of Hamiltonian paths in tournaments

Solving an old conjecture of Szele we show that the maximum number of directed Hamiltonian paths in a tournament onn vertices is at mostc · n 3/2 · n!/2 n−1, wherec is a positive constant independent ofn.

Construction of all Hamiltonian paths in tournaments

Construction of all Hamiltonian paths in tournaments

Antidirected Hamiltonian paths in tournaments

In this paper we present a short proof of Grünbaum's theorem concerning the existence of antidirected Hamiltonian (ADH) paths in tournaments. We also prove that, in every tournament T n with n ≥ 12, there is an ADH path starting at any vertex.

Antidirected Hamiltonian paths in tournaments

It is well known that every tournament has a directed Hamiltonian path. A similar result is established concerning the existence of antidirected Hamiltonian paths in tournaments.