For integers $k⩾1$ and $n⩾2k+1$, the Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\dots ,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k⩾3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form $K(2k+2^a,k)$ with $k⩾3$ and $a⩾0$ have a Hamilton cycle. We also prove that $K(2k+1,k)$ has at least $2^{2^{k-6}}$ distinct Hamilton cycles for $k⩾6$. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words.

中文翻译：

---

稀疏 Kneser 图是哈密顿量

对于整数 $克⩾1$ 和 $n⩾2克+1$, Kneser 图 $钾(n,克)$ 是顶点为 $克$-元素子集 $\{1,\dots ,n\}$并且其边连接不相交的子集对。形式的 Kneser 图$钾(2克+1,克)$也称为奇数图。由于 1970 年代的 Meredith、Lloyd 和 Biggs，我们解决了一个老问题，证明对于每个$克⩾3$，奇数图 $钾(2克+1,克)$有哈密顿循环。这和由 Johnson 引起的已知条件结果意味着所有形式的 Kneser 图$钾(2克+2^{\mathrm{一种}},克)$ 和 $克⩾3$ 和 $\mathrm{一种}⩾0$有一个汉密尔顿循环。我们也证明$钾(2克+1,克)$ 至少有 $2^{2^{克-6}}$ 不同的哈密顿循环 $克⩾6$. 我们的证明基于将奇数图中的哈密顿性问题简化为在 Dyck 词的适当定义的超图中找到生成树的问题。