A strengthening of a degree sequence condition for Hamiltonicity in tough graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909542280855552 |
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| author | Shan, Songling Tanyel, Arthur |
| author_facet | Shan, Songling Tanyel, Arthur |
| contents | Generalizing Chvátal's classic 1972 result, Hoàng proposed in 1995 the following conjecture, which strengthens Chvátal's result in terms of toughness: Let $t\ge 1$ be a positive integer and $G$ be a $t$-tough graph on $n \ge 3$ vertices with degree sequence $d_1, d_2, \dots, d_n$ in non-increasing order. Suppose for each $i\in [1, \lfloor\frac{n-1}{2} \rfloor]$, if $d_i \le i \text{ and } d_{n-i+t} < n - i $ implies $d_j + d_{n-j+t} \ge n$ for all $j\in [i+1, \lfloor\frac{n-1}{2} \rfloor]$, then $G$ is Hamiltonian. Hoàng verified the conjecture for $t=1$. In this paper, we verfity the conjecture for all $t\ge 4$. Our proof relies on a toughness closure lemma for $t\ge 4$ that we previously established. Additionally, we show that the toughness closure lemma does not hold when $t=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14735 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A strengthening of a degree sequence condition for Hamiltonicity in tough graphs Shan, Songling Tanyel, Arthur Combinatorics Generalizing Chvátal's classic 1972 result, Hoàng proposed in 1995 the following conjecture, which strengthens Chvátal's result in terms of toughness: Let $t\ge 1$ be a positive integer and $G$ be a $t$-tough graph on $n \ge 3$ vertices with degree sequence $d_1, d_2, \dots, d_n$ in non-increasing order. Suppose for each $i\in [1, \lfloor\frac{n-1}{2} \rfloor]$, if $d_i \le i \text{ and } d_{n-i+t} < n - i $ implies $d_j + d_{n-j+t} \ge n$ for all $j\in [i+1, \lfloor\frac{n-1}{2} \rfloor]$, then $G$ is Hamiltonian. Hoàng verified the conjecture for $t=1$. In this paper, we verfity the conjecture for all $t\ge 4$. Our proof relies on a toughness closure lemma for $t\ge 4$ that we previously established. Additionally, we show that the toughness closure lemma does not hold when $t=1$. |
| title | A strengthening of a degree sequence condition for Hamiltonicity in tough graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.14735 |