Oriented Hamiltonian Cycles in Tournaments

Let T be a tournament and let D be a digraph. We say that T contains D if D is a subgraph of T. The order of a digraph D, denoted by |D|, is its number of vertices. Let x and y be two vertices of T. We write x y if (x, y) is an arc of T. Likewise, let X and Y be two subdigraphs of T. We write X Y if x y for all pairs (x, y) # V(X )_V(Y ). Let A1 , A2 , ..., Ak be a family of subdigraphs of T. We denote by T(A1 , A2 , ..., Ak) the subtournament induced by T on the set of vertices 1 i k V(Ai) and by T&(A1 , A2 , ..., Ak) the subtournament induced by T on the set of vertices V(T )" 1 i k V(Ai). Let P=(x1 , ..., xn) be a path. We say that x1 is the origin of P and xn is the terminus of P. If x1 x2 , P is an outpath, otherwise P is an inpath. The directed outpath of order n is the path P=(x1 , ..., xn) in which xi xi+1 for all i, 1 i<n; the dual notion is the directed inpath; both directed outand inpaths are dipaths. The length of a path is its number of arcs. We denote the subpath (x1 , ..., xn&1) of P by P* and the subpath (x2 , ..., xn) of P by *P. The path (xn , xn&1 , ..., x1) is denoted P. doi:10.1006 jctb.2000.1959, available online at http: www.idealibrary.com on