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 pair of vertices u, v ∈ V (G), a cycle is called a geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle. A graph G is pancyclic [12] if it contains a cycle of every length from 3 to |V (G)| inclusive. Furthermore, a graph G is called geodesic k-pancyclic [3] if for each pair of vertices u, v ∈ V (G), it contains a geodesic cycle of every integer length of ...

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.

The Möbius cube Mn is a variant of the hypercube Qn and has better properties than Qn with the same number of links and processors. It has been shown by Fan [J. Fan, Hamilton-connectivity and cycle-embedding of Möbius cubes, Inform. Process. Lett. 82 (2002) 113–117] and Huang et al. [W.-T. Huang, W.-K. Chen, C.-H. Chen, Pancyclicity of Möbius cubes, in: Proc. 9th Internat. Conf. on Parallel and...

It is known that Θ(log n) chords must be added to an n-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, Θ(n) chords are required. A possibly ‘intermediate’ variation is the following: given k, 1 ≤ k ≤ n, how many chords must be added to ensure that there exist cycles of every possible length each of which passes exactly k chords...

An edge-colored graph is \emph{rainbow }if no two edges of the have same color. $G^c$ called \emph{properly colored} if every adjacent receive distinct colors in $G^c$. A \emph{strongly edge-colored} a proper such that path length $3$ rainbow. We call an vertex pair-pancyclic} any vertices are contained rainbow cycle $\ell$ for each with $3 \leq \ell n$. In this paper, we show strongly order $n...

A graph G is said to be pancyclic if G contains cycles of all lengths from 3 to |V (G)|. We show that if G is 4-connected, claw-free, and P10-free, then G is either pancyclic or it is the line graph of the Petersen graph. This implies that every 4-connected, claw-free, P9-free graph is pancyclic, which is best possible and extends a result of Gould et al. Pancyclicity in 3-connected graphs: Pai...