On the number of cycles in local tournaments

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. A vertex of a strongly connected digraph is called a non-separating vertex if its removal preserves the strong connectivity of the digraph in question. In 1990, Bang-Jensen showed that a strongly connected local tournament does not have any non-separating vertices if and only if it is a directed cycle. Guo and Volkmann extended this result in 1994. They determined the strongly connected local tournament with exactly one non-separating vertex. In the first part of this paper we characterize the class of strongly connected local tournaments with exactly two non-separating vertices. In the second part of the paper we consider the following problem: Given a strongly connected local tournament D of order n with at least n + 2 arcs and an integer 3 <= r <= n. How many directed cycles of length r exist in D? For tournaments this problem was treated by Moon in 1966 and Las Vergnas in 1975. A reformulation of the results of the first part shows that we have characterized the class of strongly connected local tournaments with exactly two directed cycles of length n - 1. Among other things we show that D has at least n - r + 1 directed cycles of length r for 4 <= r <= n - 1 unless it has a special structure. Moreover, we characterize the class of local tournaments with exactly n - r + I directed cycles of length r for 4 <= r <= n - 1 which generalizes a result of Las Vergnas. (C) 2008 Elsevier B.V. All rights reserved.