A spectral condition for Hamilton cycles in tough bipartite graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908479700074496 |
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| author | Ai, Lianyang Zhang, Wenqian |
| author_facet | Ai, Lianyang Zhang, Wenqian |
| contents | Let $G$ be a graph. The {\em spectral radius} of $G$ is the largest eigenvalue of its adjacency matrix. For a non-complete bipartite graph $G$ with parts $X$ and $Y$, the {\em bipartite toughness} of $G$ is defined as $t^{B}(G)=\min\left\{\frac{|S|}{c(G-S)}\right\}$, where the minimum is taken over all proper subsets $S\subset X$ (or $S\subset Y$) such that $c(G-S)>1$. In this paper, we give a sharp spectral radius condition for balanced bipartite graphs $G$ with $t^{B}(G)\geq1$ to guarantee that $G$ contains Hamilton cycles. This solves a problem proposed in \cite{CFL}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_03778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A spectral condition for Hamilton cycles in tough bipartite graphs Ai, Lianyang Zhang, Wenqian Combinatorics Let $G$ be a graph. The {\em spectral radius} of $G$ is the largest eigenvalue of its adjacency matrix. For a non-complete bipartite graph $G$ with parts $X$ and $Y$, the {\em bipartite toughness} of $G$ is defined as $t^{B}(G)=\min\left\{\frac{|S|}{c(G-S)}\right\}$, where the minimum is taken over all proper subsets $S\subset X$ (or $S\subset Y$) such that $c(G-S)>1$. In this paper, we give a sharp spectral radius condition for balanced bipartite graphs $G$ with $t^{B}(G)\geq1$ to guarantee that $G$ contains Hamilton cycles. This solves a problem proposed in \cite{CFL}. |
| title | A spectral condition for Hamilton cycles in tough bipartite graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.03778 |