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

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 DD of order nn and an integer 3≤r≤n3≤r≤n, how many directed cycles of length rr exist in DD?Bang-Jensen [1] showed in 1990 that every strongly connected local tournament has a directed Hamiltonian cycle, thus solving the case r=nr=n. In 2009, Meierling and Volkmann [8] showed that a strongly connected local tournament DD has at least n−r+1n−r+1 directed cycles of length rr for 4≤r≤n−14≤r≤n−1 unless it has a special structure.In this paper, we investigate the case r=3r=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=3r=3 and r=nr=n, respectively.