Bipartite holes, degree sums and Hamilton cycles
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915592041136128 |
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| author | Ellingham, Mark Huang, Yixuan Wei, Bing |
| author_facet | Ellingham, Mark Huang, Yixuan Wei, Bing |
| contents | The {\em bipartite-hole-number} of a graph $G$, denoted as $\widetildeα(G)$, is the minimum number $k$ such that there exist integers $a$ and $b$ with $a + b = k+1$ such that for any two disjoint sets $A, B \subseteq V(G)$, there is an edge between $A$ and $B$.
McDiarmid and Yolov initiated research on bipartite holes by extending Dirac's classical theorem on minimum degree and Hamiltonian cycles.
They showed that a graph on at least three vertices with $δ(G) \ge \widetildeα(G)$ is Hamiltonian.
Later, Draganić, Munhá Correia and Sudakov proved that $δ\ge \widetildeα(G)$ implies that $G$ is pancyclic, unless $G = K_{\frac n2, \frac n2}$. This extended the result of McDiarmid and Yolov and generalized a theorem of Bondy on pancyclicity.
In this paper, we show that a $2$-connected graph $G$ is Hamiltonian if $σ_2(G) \ge 2 \widetildeα(G) - 1$, and that a connected graph $G$ contains a cycle through all vertices of degree at least $\widetildeα(G)$.
Both results extended McDiarmid and Yolov's result.
As a step toward proving pancyclicity, we show that if an $n$-vertex graph $G$ satisfies $σ_2(G) \ge 2 \widetildeα(G) - 1$, then it either contains a triangle or it is $K_{\frac n2, \frac n2}$.
Finally, we discuss the relationship between connectivity and the bipartite hole number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00616 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bipartite holes, degree sums and Hamilton cycles Ellingham, Mark Huang, Yixuan Wei, Bing Combinatorics Primary: 05C45, Secondary: 05C07, 05C38, 05C40 The {\em bipartite-hole-number} of a graph $G$, denoted as $\widetildeα(G)$, is the minimum number $k$ such that there exist integers $a$ and $b$ with $a + b = k+1$ such that for any two disjoint sets $A, B \subseteq V(G)$, there is an edge between $A$ and $B$. McDiarmid and Yolov initiated research on bipartite holes by extending Dirac's classical theorem on minimum degree and Hamiltonian cycles. They showed that a graph on at least three vertices with $δ(G) \ge \widetildeα(G)$ is Hamiltonian. Later, Draganić, Munhá Correia and Sudakov proved that $δ\ge \widetildeα(G)$ implies that $G$ is pancyclic, unless $G = K_{\frac n2, \frac n2}$. This extended the result of McDiarmid and Yolov and generalized a theorem of Bondy on pancyclicity. In this paper, we show that a $2$-connected graph $G$ is Hamiltonian if $σ_2(G) \ge 2 \widetildeα(G) - 1$, and that a connected graph $G$ contains a cycle through all vertices of degree at least $\widetildeα(G)$. Both results extended McDiarmid and Yolov's result. As a step toward proving pancyclicity, we show that if an $n$-vertex graph $G$ satisfies $σ_2(G) \ge 2 \widetildeα(G) - 1$, then it either contains a triangle or it is $K_{\frac n2, \frac n2}$. Finally, we discuss the relationship between connectivity and the bipartite hole number. |
| title | Bipartite holes, degree sums and Hamilton cycles |
| topic | Combinatorics Primary: 05C45, Secondary: 05C07, 05C38, 05C40 |
| url | https://arxiv.org/abs/2511.00616 |