An Ore-type condition for hamiltonicity in graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915231422218240 |
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| author | Li, Chengli Liu, Feng |
| author_facet | Li, Chengli Liu, Feng |
| contents | The bipartite-hole-number of a graph $G$, denoted as $\widetildeα(G)$, 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, based on Ore-type conditions, we show that if a graph $G$ is 2-connected and the degree sum of any two nonadjacent vertices in $G$ is at least $ 2\widetildeα(G)$, then $G$ is hamiltonian. Furthermore, we prove that if $G$ is 3-connected and the degree sum of any two nonadjacent vertices in $G$ is at least $ 2\widetildeα(G)+1$, then $G$ is hamiltonian-connected. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04493 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Ore-type condition for hamiltonicity in graphs Li, Chengli Liu, Feng Combinatorics The bipartite-hole-number of a graph $G$, denoted as $\widetildeα(G)$, 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, based on Ore-type conditions, we show that if a graph $G$ is 2-connected and the degree sum of any two nonadjacent vertices in $G$ is at least $ 2\widetildeα(G)$, then $G$ is hamiltonian. Furthermore, we prove that if $G$ is 3-connected and the degree sum of any two nonadjacent vertices in $G$ is at least $ 2\widetildeα(G)+1$, then $G$ is hamiltonian-connected. |
| title | An Ore-type condition for hamiltonicity in graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.04493 |