Let f(n) be the smallest integer such that for every graph G of order n with minimum degree 3(G)>f(n), the line graph L(G) of G is pancyclic whenever L(G) is hamiltonian. Results are proved showing that f(n) = ®(n 1/3).

For a graph G of order n we consider the unique partition of its vertex set V (G) = A ∪ B with A = {v ∈ V (G) : d(v) ≥ n/2} and B = {v ∈ V (G) : d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.

A graph G of order n is said to be in the class O(n1) if deg(u)+deg(v)>n1 for every pair of nonadjacent vertices II, VE V(G). We characterise the graphs in O(n1) which are pancyclic.

We prove that a k-ary 2-cube Q2 with 3 faulty edges but where every vertex is incident with at least 2 healthy edges is bipancyclic, if k ≥ 3, and k-pancyclic, if k ≥ 5 is odd (these results are optimal).

This is intended as a survey article covering recent developments in the area of hamiltonian graphs, that is, graphs containing a spanning cycle. This article also contains some material on related topics such as traceable, hamiltonian-connected and pancyclic graphs and digraphs, as well as an extensive bibliography of papers in the area. 1. Supported by O.N.R. contract number N000014-88-K-0070.

Schmeichel and Hakimi ?], and Bauer and Schmeichel ?] gave an evidence in support of the well-known Bondy's "metaconjecture" that almost any non-trivial condition on graphs which implies that the graph is hamiltonian also implies that it is pancyclic. In particular, they proved that the metaconjecture is valid for Chvv atal's condition ?]. We slightly generalize their results giving a new Chvv ...

We study two topological properties of the 5-ary n-cube Qn. Given two arbitrary distinct nodes x and y in Q 5 n, we prove that there exists an x-y path of every length ranging from 2n to 5n−1, where n ≥ 2. Based on this result, we prove that Qn is 5-edge-pancyclic by showing that every edge in Qn lies on a cycle of every length ranging from 5 to 5. Mathematics Subject Classification. 68R10, 68R...

We call a graph G pancyclic if it contains at least one cycle of every possible length m, for 3 ≤ m ≤ |V (G)|. In this paper, we define a new property called chorded pancyclicity. We explore forbidden subgraphs in claw-free graphs sufficient to imply that the graph contains at least one chorded cycle of every possible length 4, 5, . . . , |V (G)|. In particular, certain paths and triangles with...

Let D be an oriented graph of order n 9, minimum degree n ? 2, such for choice of distinct vertices x and y, either xy 2 E(D) or d + (x)+d ? (y) n?3. Song (J. Graph Theory 18 (1994), 461{468) proved that D is pancyclic. In this note, we give a short proof, based on Song's result, that D is in fact vertex pancyclic. This also generalizes a result of Jackson (J. Graph Theory 5 (1981), 147{157) fo...