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

Abstract

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.