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Forbidden Pairs and (k,m)-Pancyclicity

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Crane C. B.

Discussiones Mathematicae Graph Theory

2017

tom 37

nr 3

649-663

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.

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1 Discussiones Mathematicae Graph Theory

1 Crane C. B.

1 2017

Wyniki wyszukiwania

Forbidden Pairs and (k,m)-Pancyclicity