## Abstract

We adopt the recently introduced concept of the bipartite-hole-number due to McDiarmid and Yolov, and extend their result on Hamiltonicity to other Hamiltonian properties of graphs with a large minimum degree in terms of this concept. An (s, t)-bipartite-hole in a graph G consists of two disjoint sets of vertices S and T with |S| = s and |T| = t such that E(S, T) = θ. The bipartite-hole-number ∼α(G) is the maximum integer r such that G contains an (s, t)-bipartite-hole for every pair of nonnegative integers s and t with s + t = r. Our main results are that a graph G is traceable if δ(G) ≥ ∼α(G)-1, and Hamilton-connected if δ(G) ≥ ∼;α(G) + 1, both improving the analogues of Dirac's Theorem for traceable and Hamilton-connected graphs.

Original language | English |
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Pages (from-to) | 717-726 |

Number of pages | 10 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 44 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2024 |

## Keywords

- Bipartite-hole-number
- Degree condition
- Hamilton-connected graph
- Minimum degree.
- Traceable graph