Geodesic
Pancyclism and small cycles in graphs
Discussiones Mathematicae. Graph Theory, Tome 16 (1996) no. 1, pp. 27-40

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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 for each 3 ≤ m ≤ max (d_C(u,v)+1, [(n+19)/13]), d_C(u,v) being the distance of u and v on a hamiltonian cycle of G.
Keywords: cycle, hamiltonian, pancyclic 
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Faudree, Ralph; Favaron, Odile; Flandrin, Evelynei; Li, Hao. Pancyclism and small cycles in graphs. Discussiones Mathematicae. Graph Theory, Tome 16 (1996) no. 1, pp. 27-40. http://geodesic.mathdoc.fr/item/DMGT_1996_16_1_a2/