Dirac-type condition for Hamilton-generated graphs
Xinmin Hou, Zhi Yin
TL;DR
This work resolves a Dirac-type problem for Hamilton-generated graphs by showing that for sufficiently large odd $n$, every Hamilton-connected graph on $n$ vertices with minimum degree at least $\frac{n-1}{2}$ satisfies $\mathcal{C}_{n}(G)=\mathcal{C}(G)$, i.e., every cycle is a symmetric difference of Hamilton cycles. The authors deploy the parity-switcher technique (from recent work of Christoph, Nenadov, and Petrova) together with a robust structural classification to rule out all potential obstructions to Hamilton-generatedness. The result is tight in both the minimum-degree bound and the Hamilton-connected condition, with constructive examples illustrating the limits. This advances Bondy-type questions under Dirac-type thresholds and suggests the parity-switcher framework as a powerful tool for cycle-space generation problems in dense graphs.
Abstract
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).