An arc in a tournament T with n ≥ 3 vertices is called k-pancyclic, if it belongs to a cycle of length l for all k ≤ l ≤ n. In this paper, the result that each s-strong (s ≥ 3) tournament T contains at least s + 2 out-arc 5-pancyclic vertices is obtained. Furthermore, our proof yields a polynomial algorithm to find s + 2 out-arc 5-pancyclic vertices of T .

Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 36 k6 n, and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 36 k6 n. In this paper, we shall present di0erent su1cient conditions for graphs to be vertex pancyclic. ? 2002 Elsevier Science B.V. All rights reserved.

Abstract: A hypertournament or a k-tournament, on n vertices, 2≤k≤n, is a pair T= (V,E), where the vertex setV is a set of size n and the edge setE is the collection of all possible subsets of size k of V, called the edges, each taken in one of its k! possible permutations. A k-tournament is pancyclic if there exists (directed) cycles of all possible lengths; it is vertex-pancyclic if moreover ...

Say that a cycle C almost contains a cycle C− if every edge except one of C− is an edge of C. Call a graph G strongly pancyclic if every nontriangular cycle C almost contains another cycle C− and every nonspanning cycle C is almost contained in another cycle C. This is equivalent to requiring, in addition, that the sizes of C− and C differ by one from the size of C. Strongly pancyclic graphs ar...

A graph G is pancyclic if it contains cycles of each length `, 3 ≤ ` ≤ |V (G)|. The generalized bull B(i, j) is obtained by associating one endpoint of each of the paths Pi+1 and Pj+1 with distinct vertices of a triangle. Gould, Luczak and Pfender [4] showed that if G is a 3-connected {K1,3, B(i, j)}-free graph with i + j = 4 then G is pancyclic. In this paper, we prove that every 4-connected, ...