Geodesic
Forbidden Pairs and (k, m)-Pancyclicity
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 3, pp. 649-663

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A graph G on n vertices is said to be (k,m)-pancyclic if every set of k vertices in G is contained in a cycle of length r for each r ∈ m, m+1, . . ., n. This property, which generalizes the notion of a vertex pancyclic graph, was defined by Faudree, Gould, Jacobson, and Lesniak in 2004. The notion of (k, m)-pancyclicity provides one way to measure the prevalence of cycles in a graph. We consider pairs of subgraphs that, when forbidden, guarantee hamiltonicity for 2-connected graphs on n ≥ 10 vertices. There are exactly ten such pairs. For each integer k ≥ 1 and each of eight such subgraph pairs R, S, we determine the smallest value m such that any 2-connected R, S-free graph on n ≥ 10 vertices is guaranteed to be (k,m)-pancyclic. Examples are provided that show the given values are best possible. Each such example we provide represents an infinite family of graphs.
Keywords: hamiltonian, pancyclic, forbidden subgraph, cycle, claw-free 
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     author = {Crane, Charles Brian},
     title = {Forbidden {Pairs} and (k, {m)-Pancyclicity}},
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Crane, Charles Brian. Forbidden Pairs and (k, m)-Pancyclicity. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 3, pp. 649-663. http://geodesic.mathdoc.fr/item/DMGT_2017_37_3_a11/