Given integers k, s, t with 0 ≤ s ≤ t and k ≥ 0, a (k, t, s)-linear forest F is a graph that is the vertex disjoint union of t paths with a total of k edges and with s of the paths being single vertices. If the number of single vertex paths is not critical, the forest F will simply be called a (k, t)-linear forest. A graph G of order n ≥ k + t is (k, t)-hamiltonian if for any (k, t)-linear fore...

Tewes and Volkmann [Australas. J. Combin. 18 (1998), 293–301] proved that for an integer k ≥ 3, every in-tournament of minimum in-degree d ≥ 1 and of order at most kd contains a directed cycle of length at most k. In other words, they proved that the Caccetta-Häggkvist conjecture, with respect to the minimum in-degree, is true for in-tournaments. In the same paper, they proved also that every s...

We show that for every 3-connected cubic graph G, the prism G×K2 has cycles of every even length. Furthermore, if G has a triangle, then G × K2 is pancyclic.

In this paper, we study the existence of cycles of all lengths in the recursive circulant graphs, and we show a necessary and sufficient condition for the graph being pancyclic and bipancyclic.  2002 Elsevier Science B.V. All rights reserved.

Abstract Motivated by several conjectures due to Nikoghosyan, in a recent article Li et al., the aim was characterize all possible graphs H such that every 1-tough -free graph is hamiltonian. The almost complete answer given there conclusion proper induced subgraph of $$K_1\cup P_4$$ K 1</mm...

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

A graph G on n vertices is pancyclic if G contains cycles of all lengths`for 3 ` n and G is cycle extendable if for every nonhamiltonian cycle C G there is a cycle C 0 G such that V (C) V (C 0) and jV (C 0) n V (C)j = 1. We prove that (i) every 2-connected K 1;3-free graph is pancyclic, if G is P 5-free and n 6, if G is P 6-free and n 10, or if G is P 7-free, deer-free and n 14, and (ii) every ...