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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2503.15950 |
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| _version_ | 1866909545459089408 |
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| author | Hou, Xinmin Yin, Zhi |
| author_facet | Hou, Xinmin Yin, Zhi |
| contents | The cycle space $\mathcal{C}(G)$ of a graph $G$ is defined as the linear space spanned by all cycles in $G$. For an integer $k\ge 3$, let $\mathcal{C}_k (G)$ denote the subspace of $\mathcal{C}(G)$ generated by the cycles of length exactly $k$. A graph $G$ on $n$ vertices is called Hamilton-generated if $\mathcal{C}_n (G) = \mathcal{C}(G)$, meaning every cycle in $G$ is a symmetric difference of some Hamilton cycles of $G$. %A necessary condition for this property is that $n$ must be odd. Heinig (European J. Combin., 2014) showed that for any $σ>0$ and sufficiently large odd $n$, every $n$-vertex graph with minimum degree $(1+ σ)n/2$ is Hamilton-generated. He further posed the question that whether the minimum degree requirement could be lowered to the Dirac threshold $n/2$. Recent progress by Christoph, Nenadov, and Petrova~(arXiv:2402.01447) reduced the minimum degree condition to $n/2 + C$ for some large constant $C$. In this paper, we resolve Heinig's problem completely by proving that for sufficiently large odd $n$, every Hamilton-connected graph $G$ on $n$ vertices with minimum degree at least $(n-1)/2$ is Hamilton-generated. Moreover, this result is tight for the minimum degree and the Hamilton-connected condition. The proof relies on the parity-switcher technique introduced by Christoph, et al in their recent work, as well as a classification lemma that strengthens a previous result by Krivelevich, Lee, and Sudakov~(Trans. Amer. Math. Soc., 2014). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_15950 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dirac-type condition for Hamilton-generated graphs Hou, Xinmin Yin, Zhi Combinatorics The cycle space $\mathcal{C}(G)$ of a graph $G$ is defined as the linear space spanned by all cycles in $G$. For an integer $k\ge 3$, let $\mathcal{C}_k (G)$ denote the subspace of $\mathcal{C}(G)$ generated by the cycles of length exactly $k$. A graph $G$ on $n$ vertices is called Hamilton-generated if $\mathcal{C}_n (G) = \mathcal{C}(G)$, meaning every cycle in $G$ is a symmetric difference of some Hamilton cycles of $G$. %A necessary condition for this property is that $n$ must be odd. Heinig (European J. Combin., 2014) showed that for any $σ>0$ and sufficiently large odd $n$, every $n$-vertex graph with minimum degree $(1+ σ)n/2$ is Hamilton-generated. He further posed the question that whether the minimum degree requirement could be lowered to the Dirac threshold $n/2$. Recent progress by Christoph, Nenadov, and Petrova~(arXiv:2402.01447) reduced the minimum degree condition to $n/2 + C$ for some large constant $C$. In this paper, we resolve Heinig's problem completely by proving that for sufficiently large odd $n$, every Hamilton-connected graph $G$ on $n$ vertices with minimum degree at least $(n-1)/2$ is Hamilton-generated. Moreover, this result is tight for the minimum degree and the Hamilton-connected condition. The proof relies on the parity-switcher technique introduced by Christoph, et al in their recent work, as well as a classification lemma that strengthens a previous result by Krivelevich, Lee, and Sudakov~(Trans. Amer. Math. Soc., 2014). |
| title | Dirac-type condition for Hamilton-generated graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.15950 |