A graph G on n vertices is said to be pancyclic if it contains cycles of all lengths k for k ∈ {3, . . . , n}. A vertex v ∈ V (G) is called super-heavy if the number of its neighbours in G is at least (n+1)/2. For a given graph H we say that G is H-f1-heavy if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), dK(u, v) = 2 implies that at least one of them is super-heavy. For a family of graphs H we say that G is H-f1-heavy, if G is H-f1-heavy for every graph H ∈H. Let D denote the deer, a graph consisting of a triangle with two disjoint paths P3 adjoined to two of its vertices. In this paper we prove that every 2-connected {K1,3, P7, D}-f1-heavy graph on n ≥ 14 vertices is pancyclic. This result extends the previous work by Faudree, Ryjáček and Schiermeyer.
@article{bwmeta1.element.doi-10_7151_dmgt_1938,
author = {Wojciech Wide},
title = {A Triple of Heavy Subgraphs Ensuring Pancyclicity of 2-Connected Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {477-499},
zbl = {06705141},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1938}
}
Wojciech Wide. A Triple of Heavy Subgraphs Ensuring Pancyclicity of 2-Connected Graphs. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 477-499. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1938/