Let C denote the claw , N the net (a graph obtained from a K₃ by attaching a disjoint edge to each vertex of the K₃), W the wounded (a graph obtained from a K₃ by attaching an edge to one vertex and a disjoint path P₃ to a second vertex), and the graph consisting of a K₃ with a path of length i attached to one vertex. For k a fixed positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY-free (does not contain an induced copy of C or of Y) will be determined for Y a connected subgraph of either P₆, N, W, or Z₃. It should be noted that the pairs of graphs CY are precisely those forbidden pairs that imply that any 2-connected graph of order at least 10 is hamiltonian. These extremal numbers give one measure of the relative strengths of the forbidden subgraph conditions that imply a graph is hamiltonian.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1082,
author = {Ralph Faudree and Andr\'as Gy\'arf\'as},
title = {Extremal problems for forbidden pairs that imply hamiltonicity},
journal = {Discussiones Mathematicae Graph Theory},
volume = {19},
year = {1999},
pages = {13-29},
zbl = {0927.05039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1082}
}
Ralph Faudree; András Gyárfás. Extremal problems for forbidden pairs that imply hamiltonicity. Discussiones Mathematicae Graph Theory, Tome 19 (1999) pp. 13-29. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1082/
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