### Abstract

Let n,d be integers with 1 ≤ d ≤ [n-1/2], and set h(n,d):= (n-d/2)+ d^{2}. Erdős proved that when n ≥ 6d, each n-vertex nonhamiltonian graph G with minimum degree δ(G) ≥ d has at most h(n,d) edges. He also provides a sharpness example H_{n,d} for all such pairs n,d. Previously, we showed a stability version of this result: for n large enough, every nonhamiltonian graph G on n vertices with δ(G) ≥ d and more than h(n,d+1) edges is a subgraph of H_{n,d} . In this article, we show that not only does the graph H_{n,d} maximize the number of edges among nonhamiltonian graphs with n vertices and minimum degree at least d, but in fact it maximizes the number of copies of any fixed graph F when n is sufficiently large in comparison with d and (Formula presented.). We also show a stronger stability theorem, that is, we classify all nonhamiltonian n-vertex graphs with δ(G) ≥ d and more than h(n,d +2) edges. We show this by proving a more general theorem: we describe all such graphs with more than (n-(d+2)/k)+ (d+2) (d+2 k-1) copies of K_{k} for any k.

Original language | English |
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Pages (from-to) | 176-193 |

Number of pages | 18 |

Journal | Journal of Graph Theory |

Volume | 89 |

Issue number | 2 |

DOIs | |

Publication status | Published - Oct 2018 |

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### Keywords

- extremal graph theory
- hamiltonian cycles
- subgraph density

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*89*(2), 176-193. https://doi.org/10.1002/jgt.22246