We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u) + d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u) + d(v) ≥ n + z, where z = 0 when n is odd and z = 1 otherwise, then G contains a cycle of length m ...

Let G be a 2-connected graph with n ?: 3 vertices such that for any two vertices u, v at distance two in an induced subgraph K 1,3 or P4 of G, the inequality d(u) + d(v) ?: IN(u) U N(v) U N(w)ls holds for all w E N(u) n N(v). We prove that (i) if s = 1 and IN(u) n N(v)1 ?: 2, then G is hamiltonian or K p,p+1 ~ G ~ Kp + K p+1; (ii) if s = 0, then G is either pancyclic, or bipartite graph. This g...

The coupled graph c(G) of a plane graph G is the graph de.ned on the vertex set V (G)∪F(G) so that two vertices in c(G) are joined by an edge if and only if they are adjacent or incident in G. We prove that the coupled graph of a 2-connected plane graph is edge-pancyclic. However, there exists a 2-edge-connected plane graph G such that c(G) is not Hamiltonian. ? 2002 Elsevier Science B.V. All r...

Twisted cubes, crossed cubes, Möbius cubes, and locally twisted cubes are some of the widely studied hypercube variants. The 4-pancyclicity of twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes and the 4-edgepancyclicity of crossed cubes are proven in [2, 1, 5, 10, 6] respectively. It should be noted that 4-edge-pancyclicity implies 4-node-pancyclicity which further implies 4-pan...

A k-hypertournament H on n vertices, where 2 ≤ k ≤ n, is a pair H = (V,AH), where V is the vertex set of H and AH is a set of k-tuples of vertices, called arcs, such that for all subsets S ⊆ V of order k, AH contains exactly one permutation of S as an arc. Inspired by the successful extension of classical results for tournaments (i.e. 2-hypertournaments) to hypertournaments, by Gutin and Yeo [J...

Let G be a graph and H be a subgraph of G. If G contains a hamiltonian cycle C such that E(C)∩E(H) is empty, we say that C is an H-avoiding hamiltonian cycle. Let F be any graph. If G contains an H-avoiding hamiltonian cycle for every subgraph H of G such that H ∼= F , then we say that G is F -avoiding hamiltonian. In this paper, we give minimum degree and degree-sum conditions which ensure tha...

The OTIS-Network (also referred to as two-level swapped network) is composed of n clones of an n-node original network constituting its clusters. It has received much attention due to its many favorable properties such as high degree of scalability, regularity, modularity, package-ability and high degree of algorithmic efficiency. In this paper, using the construction method, we show that the O...