Local tournaments with the minimum number of Hamiltonian cycles or cycles of length three

• Published in: Discrete mathematics Vol. 310; no. 13; pp. 1940 - 1948
• Main Author: Meierling, Dirk
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 28.07.2010; Elsevier
• Subjects: Algebra; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Graph theory; Integers; Local tournament; Lower bounds; Mathematical analysis; Mathematics; Number of cycles; Sciences and techniques of general use; Number of cycles; Local tournament; Lower bound; Vertex; Hamiltonian cycle; Solving; N order; Integer; Discrete mathematics; Combinatorics; Digraph; Tournament; Directed graph
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2010.03.003

A digraph without loops, multiple arcs and directed cycles of length two is called a local tournament if the set of in-neighbors as well as the set of out-neighbors of every vertex induces a tournament. In this paper we consider the following problem: Given a strongly connected local tournament D of order n and an integer 3 ≤ r ≤ n , how many directed cycles of length r exist in D ? Bang-Jensen  [1] showed in 1990 that every strongly connected local tournament has a directed Hamiltonian cycle, thus solving the case r = n . In 2009, Meierling and Volkmann  [8] showed that a strongly connected local tournament D has at least n − r + 1 directed cycles of length r for 4 ≤ r ≤ n − 1 unless it has a special structure. In this paper, we investigate the case r = 3 and present a lower bound for the number of directed cycles of length three. Furthermore, we characterize the classes of local tournaments achieving equality in the bounds for r = 3 and r = n , respectively.