发表机构
Sungkyunkwan University; The State University of New York, Korea（成均馆大学; 纽约州立大学韩国分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了大阶数下$3$-图和$q$-平衡$3$部$3$-图中基于局部谱半径的完美匹配猜想，通过分数匹配稳定性定理识别空间障碍。
AI 中文摘要
对于一个$3$-均匀超图$H$，设$\sigma(H)$为$H$的顶点上链接谱半径的最小值。Lin、Lu、Yuan和Zhao猜想，若$\sigma(H)>\frac{2n}{3}-2$，则在一个阶数$n$能被3整除的$3$-图中存在完美匹配；Lu和Yuan猜想，若$\sigma(H)>\tau(q)$，则在一个$q$-平衡的$3$部$3$-图中存在完美匹配，其中对于奇数$q$，$\tau(q)=\sqrt{q(q-1)/2}$，对于偶数$q$，$\tau(q)$由一个四次方程给出。在本文中，我们证明了大阶数下这两个完美匹配猜想。每个证明都通过一个关于分数匹配的稳定性定理：我们得到一个具有零坐标的权重不足的分数顶点覆盖，该顶点的链接承载诱导覆盖，并且该链接的谱半径由一个不等式界定，其极值情形识别出相应的空间障碍。
英文摘要
For a $3$-uniform hypergraph $H$, let $σ(H)$ be the minimum over the vertices of $H$ of the spectral radius of the link. Lin, Lu, Yuan and Zhao conjectured that $σ(H)>\tfrac{2n}3-2$ forces a perfect matching in a $3$-graph of order $n$ divisible by three, and Lu and Yuan conjectured that $σ(H)>τ(q)$ forces one in a $q$-balanced $3$-partite $3$-graph, where $τ(q)=\sqrt{q(q-1)/2}$ for odd $q$ and $τ(q)$ is given by a quartic for even $q$. In this paper, we prove both perfect matching conjectures for large order. Each proof passes through a stability theorem for fractional matchings: a fractional vertex cover of deficient weight with a zero coordinate is obtained, the link of that vertex carries the induced cover, and the spectral radius of such a link is bounded by an inequality whose extremal cases identify the corresponding space barriers.
Comments39 pages
