A graph on n vertices is called pancyclic if it contains a cycle of length ` for all 3 ≤ ` ≤ n. In 1972, Erdős proved that if G is a Hamiltonian graph on n > 4k vertices with independence number k, then G is pancyclic. He then suggested that n = Ω(k) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such th...

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...

PURPOSE To retrospectively evaluate the effectiveness of paired, arcuate transverse keratotomy (Arc-T) performed prior to or after photorefractive keratectomy (PRK) to correct low to moderate amounts of natural or laser-induced astigmatism. METHODS Spherical PRK was performed in 730 eyes for myopia of -1.00 to -7.00 D. PRK with arcuate transverse keratotomy was performed in 150 of these eyes;...

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...