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.
@article{bwmeta1.element.doi-10_7151_dmgt_1949,
author = {Charles Brian Crane},
title = {Forbidden Pairs and (k,m)-Pancyclicity},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {649-663},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1949}
}
Charles Brian Crane. Forbidden Pairs and (k,m)-Pancyclicity. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 649-663. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1949/