Antidirected spanning closed trail in tournaments

Abstract

Let D be a digraph with vertex set V(D) and arc set A(D). An antidirected spanning closed trail of D is a spanning closed trail in which consecutive arcs have opposite directions and each arc of D occurs at most once. If all vertices in this antidirected spanning closed trail are distinct, then the antidirected spanning closed trail is called an antidirected hamiltonian cycle. Grünbaum in 1971 conjectured that every even tournament Tn with n≥10 has an antidirected hamiltonian cycle. In 1983, Petrović proved that any even tournament with at least 16 vertices has an antidirected hamiltonian cycle, which was the best result supporting this conjecture by far. In this paper, we investigate the existence of antidirected spanning closed trails in tournaments. Moreover, we show that every tournament Tn with n≥6 has an antidirected spanning closed trail, and prove that it is optimal.