Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.16099 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912720501080064 |
|---|---|
| author | Cheng, Kun Tang, Yurui |
| author_facet | Cheng, Kun Tang, Yurui |
| contents | The bipartite-hole-number of a graph $G$, denoted by $\widetildeα(G)$, is the minimum number $k$ such that there exist positive integers $s$ and $t$ with $s+t=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$ with $|A|=s$ and $|B|=t$, there is an edge between $A$ and $B$. In this paper, we first prove that any $2$-connected graph $G$ satisfying $d_G(x)+d_G(y)\ge 2\widetildeα(G)-2$ for every pair of non-adjacent vertices $x,y$ is hamiltonian except for a special family of graphs, thereby extending results of Li and Liu (2025), and Ellingham, Huang and Wei (2025). We then establish a stability version of a theorem by McDiarmid and Yolov (2017): every graph whose minimum degree is at least its bipartite-hole-number minus one is hamiltonian except for a special family of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16099 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extending two results on hamiltonian graphs involving the bipartite-hole-number Cheng, Kun Tang, Yurui Combinatorics The bipartite-hole-number of a graph $G$, denoted by $\widetildeα(G)$, is the minimum number $k$ such that there exist positive integers $s$ and $t$ with $s+t=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$ with $|A|=s$ and $|B|=t$, there is an edge between $A$ and $B$. In this paper, we first prove that any $2$-connected graph $G$ satisfying $d_G(x)+d_G(y)\ge 2\widetildeα(G)-2$ for every pair of non-adjacent vertices $x,y$ is hamiltonian except for a special family of graphs, thereby extending results of Li and Liu (2025), and Ellingham, Huang and Wei (2025). We then establish a stability version of a theorem by McDiarmid and Yolov (2017): every graph whose minimum degree is at least its bipartite-hole-number minus one is hamiltonian except for a special family of graphs. |
| title | Extending two results on hamiltonian graphs involving the bipartite-hole-number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.16099 |