On king–serf pair in tournaments

Abstract

For a directed graph G, a vertex x is a king if every other vertex can be reached from x by a directed path of length at most 2 and is a serf if x can be reached from every other vertex by a directed path of length at most 2. A tournament T with vertex set V is called X-hamiltonian crossing if every hamiltonian path must have one end in X and the other in X̄=V−X. Our main results are the following: •A tournament T≠T4 is X-hamiltonian crossing if and only if T has a strong component decomposition V=V1∪⋯∪Vk such that V1⊆X and Vk⊆X̄, or V1⊆X̄ and Vk⊆X.•Let T be a tournament and (x,y) a king–serf pair in T−xy. Then there exists no hamiltonian (x,y)-path in T if and only if T is an x→y special tournament with U(S1;D1,D2)≠0̸ and U(S2;D1,D2)≠0̸.•Let T be a tournament of order n≥3. If x is a vertex of maximum out-degree and y a vertex of maximum in-degree then there exists a hamiltonian (x,y)-path if and only if xy is not exceptional.