A graph G 1⁄4 ðV ;EÞ is vertex-pancyclic if for every vertex u and any integer l ranging from a positive constant L to jV j, G contains a cycle C of length l such that u is in C. In this paper, we show that an n-dimensional augmented cube, where n P 2, is vertex-pancyclic with L 1⁄4 3. 2007 Elsevier Inc. All rights reserved.

Let G be a graph on n/> 3 vertices. Then G is vertex pancyclic if every vertex of G is contained in cycles of length 3, 4, . . . , n. Let m n denote the minimum number of edges of a vertex pancyclic graph on n vertices. We show that m 3 = 3, m, = 5, m 5 = 7, m 6 = 9, and ~ n < m. ~< [_~ n ] (n >~ 7).

We prove the following theorem. Let G be a graph of order n and let W V(G). If |W | 3 and dG(x)+dG( y) n for every pair of non-adjacent vertices x, y # W, then either G contains cycles C , C, ..., C |W | such that C i contains exactly i vertices from W (i=3, 4, ..., |W | ), or |W |=n and G=Kn 2, n 2 , or else |W |=4, G[W]=K2, 2 . This generalizes a result of J. A. Bondy (1971, J. Combin. Theory...

For two vertices X,Y ∈ V (G), a cycle is called a geodesic cycle with X and Y if a shortest path joining X and Y lies on the cycle. A graph G is called to be geodesic k-pancyclic if any two vertices X,Y on G have such geodesic cycle of length l that 2dG(X,Y ) + k ≤ l ≤ |V (G)|. In this paper, we show that the n-dimensional Möbius cube MQn is geodesic 2-pancyclic for n ≥ 3. This result is optima...

A tournament is a digraph, where there is precisely one arc between every pair of distinct vertices. An arc is pancyclic in a digraph D, if it belongs to a cycle of length l, for all 3 <= l <= |V (D)|. Let p(D) denote the number of pancyclic arcs in a digraph D and let h(D) denote the maximum number of pancyclic arcs belonging to the same Hamilton cycle of D. Note that p(D) >= h(D). Moon showed...

We study some properties of the closure concept in claw-free graphs that was introduced by the rst author. It is known that G is hamiltonian if and only if its closure is hamiltonian, but, on the other hand, there are innnite classes of non-pancyclic graphs with pancyclic closure. We show several structural properties of claw-free graphs with complete closure and their clique cutsets and, using...

A graph G is triangularly connected if for every pair of edges e1, e2 ∈ E(G), G has a sequence of 3-cycles C1, C2, · · · , Cl such that e1 ∈ C1, e2 ∈ Cl and such that E(Ci) ∩ E(Ci+1) 6= ∅, (1 ≤ i ≤ l − 1). In this paper it is shown that every triangularly connected claw-free graph G with |E(G)| ≥ 3 is vertex pancyclic. This implies the former results in [2], [3], [4] and [5] that every connecte...

A graph G on n vertices is said to be (k, m)-pancyclic if each k-set S ⊆ V (G) is contained in cycles of each of the following lengths: m,m + 1, · · · , n. This property generalizes vertex pancyclicity. There are ten pairs of forbidden subgraphs which guarantee that a 2-connected graph is (k,m)-pancyclic for some integer m ≤ n. We give the best (smallest) possible value for m in each of these t...

As an enhancement on the hypercube Qn, the augmented cube AQn, prosed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71–84], not only retains some favorable properties of Qn but also possesses some embedding properties that Qn does not. For example, AQn is pancyclic, that is, AQn contains cycles of arbitrary length for n 2. This paper shows that AQn re...