Solution of a conjecture of Volkmann on longest paths through an arc in strongly connected in‐tournaments

Abstract

An in-tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. Let m = 4 or m = 5 and let D be a strongly connected in-tournament of order n ≥ 2m − 2 such that each arc belongs to a directed path of order at least m. In 2000, Volkmann showed that if D contains an arc e such that the longest directed path through e consists of exactly m vertices, then e is the only arc of D with that property. In this article we shall see that this proposition is true for m ≥ 4, thereby validating a conjecture of Volkmann. Furthermore, we prove that if we ease the restrictions on the order of D to n ≥ 2m − 3, the in-tournament D in question has at most two such arcs. In doing so, we also give a characterization of the in-tournaments with exactly two such arcs.