Pancyclic orderings of in-tournaments Original Research Article

An in-tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. In this paper, pancyclic orderings of a strong in-tournament D are investigated. This is a labeling, say x1,x2,…,xn, of the vertex set of D such that D[{x1,x2,…,xt}] is Hamiltonian for t=3,4,…,n. Moreover, we consider the related problem on weak pancyclic orderings, where the same holds for t⩾4 and x1 belongs to an arbitrary oriented 3-cycle. We present sharp lower bounds for the minimum indegree ensuring the existence of a pancyclic or a weak pancyclic ordering in strong in-tournaments.