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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.10903 |
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| _version_ | 1866914003447447552 |
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| author | Chen, Guantao Lavrov, Mikhail Ma, Yuying Su, Yimo Vandenbussche, Jennifer |
| author_facet | Chen, Guantao Lavrov, Mikhail Ma, Yuying Su, Yimo Vandenbussche, Jennifer |
| contents | The super-neighborhood of a vertex set $A$ in a graph $G$, denoted by $Λ^2(A)$, is the set of vertices adjacent to at least two vertices in $A$. We say that a bipartite graph $G=(X, Y)$ with $|X| \geq 2$ satisfies the double Hall property (with respect to $X$) if $|Λ^2(A)| \geq |A|$ for any subset $A \subseteq X$ with $|A| \geq 2$. Kostochka et al. first conjectured that if a bipartite graph $G=(X, Y)$ satisfies a slightly weaker version of the double Hall property, then $G$ contains a cycle that covers all vertices of $X$. They verified their conjecture for $|X| \leq 6$. In this paper, we extend their result to $|X| = 7$. Later, Salia conjectured that every bipartite graph satisfying the double Hall property has a cycle covering all vertices of $X$. We show that Salia's conjecture is almost equivalent to a much weaker conjecture requiring vertices in $Y$ to have high degrees. By extending a result of Barát et al., we also show that Salia's conjecture holds for some graphs where the vertices of $Y$ have degree either $2$ or very high. Finally, we establish a lower bound for the maximum degree of graphs satisfying the double Hall property and present deterministic and probabilistic constructions of such graphs that approach this bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bipartite graphs with the double Hall property Chen, Guantao Lavrov, Mikhail Ma, Yuying Su, Yimo Vandenbussche, Jennifer Combinatorics 05C38, 05C07 The super-neighborhood of a vertex set $A$ in a graph $G$, denoted by $Λ^2(A)$, is the set of vertices adjacent to at least two vertices in $A$. We say that a bipartite graph $G=(X, Y)$ with $|X| \geq 2$ satisfies the double Hall property (with respect to $X$) if $|Λ^2(A)| \geq |A|$ for any subset $A \subseteq X$ with $|A| \geq 2$. Kostochka et al. first conjectured that if a bipartite graph $G=(X, Y)$ satisfies a slightly weaker version of the double Hall property, then $G$ contains a cycle that covers all vertices of $X$. They verified their conjecture for $|X| \leq 6$. In this paper, we extend their result to $|X| = 7$. Later, Salia conjectured that every bipartite graph satisfying the double Hall property has a cycle covering all vertices of $X$. We show that Salia's conjecture is almost equivalent to a much weaker conjecture requiring vertices in $Y$ to have high degrees. By extending a result of Barát et al., we also show that Salia's conjecture holds for some graphs where the vertices of $Y$ have degree either $2$ or very high. Finally, we establish a lower bound for the maximum degree of graphs satisfying the double Hall property and present deterministic and probabilistic constructions of such graphs that approach this bound. |
| title | Bipartite graphs with the double Hall property |
| topic | Combinatorics 05C38, 05C07 |
| url | https://arxiv.org/abs/2502.10903 |