Finding an antidirected Hamiltonian path starting with a forward arc from a given vertex of a tournament

Abstract

We prove (in two different ways) that in a sufficiently large tournament a vertex of outdegree at least two is always the beginning of an antidirected hamiltonian path starting with a forward arc. The proofs yield algorithms to find, if possible, an antidirected Hamiltonian path starting in a given vertex with an arc of a given direction. The first proof yields the theorem for all tournaments of order at least 19. The second proof only applies to somewhat larger tournaments, but leads to more efficient (sequential and parallel) algorithms.