In generalizing the concept of a pancyclic graph, we say that a graph is ‘weakly pancyclic’ if it contains cycles of every length between the length of a shortest and a longest cycle. In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic are considerably weaker than those required to ensure that it is pancyclic. This sheds some light o...

A shortest path connecting two vertices u and v is called a u-v geodesic. The distance between u and v, denoted by dG(u, v), is the number of edges in a u-v geodesic. A graph G with n vertices is geodesic-pancyclic if for each pair of vertices u, v ∈ V (G), every u-v geodesic lies on every cycle of length k satisfying max{2dG(u, v), 3} ≤ k ≤ n. In this paper, we study the properties for graphs ...

For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K1,4-free graph G with minimum degree δ(G) ≥ k+ 3 is pancyclic mod k and every claw-free graph G with δ(G) ≥ k+ 1 is pancyclic mod k, which confirms Thomassen’s conjecture [8] for claw-free graphs.

An n-vertex graph is called pancyclic if it contains a cycle of length t for all 3 ≤ t ≤ n. In this paper, we study pancyclicity of random graphs in the context of resilience, and prove that if p n−1/2, then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamiltonian subgraph of G(n, p) with more than (12 +o(1)) ( n 2 ) p edges is pancyclic. This result is best possible ...

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.

A side skirt is a planar rooted tree T, T≠P2, where the root of T vertex degree at least two, and all other vertices except leaves are three. reduced Halin graph or skirted plane G=T∪P, skirt, P path connecting in order determined by embedding T. The structure graphs contains both symmetry asymmetry. For n≥2 Pn=v1v2v3⋯vn as length n−1, we call Cartesian product G Pn, n-generalized prism over G....

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, ...