For 2 s p s n and n 2 3, D(n, p) denotes the digraph with n vertices obtained from a directed cycle of length n by changing the orientation of p -1 consecutives edges. In this paper, we prove that every tournament of order n 2 7 contains D(n, p ) for p = 2, 3, ..., n. Furthermore, we determine the tournaments of order n, 3 s n S 6, which do not contain D(n, p) for some p.A tournament T of order n is a digraph on n vertices in which every pair of distinct vertices is joined by one edge. We use the standard terminology of Berge [3]. E(T) = E denotes the set of edges and V(T) = V the set of vertices of T. A partition ( A , B ) of V , A , and B written in this order is a directed cocycle if every edge joining A and B is oriented from A to B. We know that a tournament T is strong if and only if it contains a directed Hamiltonian cycle and T is not strong if and only if it contains a directed cocycle. Let n, p be two integers, n 5 3, 2 S p S n. Then D(n, p ) = [alr a,, ..., a,; a l , h l , b2, ..., bn-,, a,] is the digraph composed of a directed path ( a l , a z , ..., a,) of length p -1 and a directed path ( a l , h , , h 2 , ..., b,,-,, a,) of length np + 1, where a, # h,, for all i , j .Griinbaum [4] and Thomassen [S, 91 have studied the problem of the existence of D(n, 2) in a tournament. Griinbaum proved that every tournament T of order n 2 3 contains D(n, 2) unless T is isomorphic to T: or TT (see Fig. 1). Thomassen proved that the minimum number of copies of D(n, 2) in a strong tournament is at least n -5 , and at least cn for some constant c > 1.

Clark proved that L(G) is hamiltonian if G is a connected graph of ordern 2 6 such that deg u + deg v 2 n -1 -p(n) for every edge uv of G, where p(n) = 0 if n is even and p(n) = 1 if n is odd. Here it is shown that the bound n -1 -dn) can be decreased to (2n + 1)/3 if every bridge of G is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of L(G). Moreover, the conclusion that L(G) is hamiltonian can be strengthened to the conclusion that L ( G ) is pancyclic. Lesniak-Foster and Williamson proved that G contains a spanning closed trail if IV(G)J = n 2 6, 6 ( G ) 2 2 and deg u + deg v 2 n -1 for every pair of nonadjacent vertices u and v. The bound n -1 can be decreased to (2n + 3)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail.

No abstract

No abstract