发表机构
Shandong University; University of Southern Denmark（山东大学; 南丹麦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了一个统一定理，给出混合刚性生成子图填充的尖锐连通性阈值，并改进了高度连通定向的二次上界，解决了两个猜想并降低了首项系数。
AI 中文摘要
Garamvölgyi、Jordán、Király和Villányi [{{\bf Forum Math. Pi} \textbf{13} (2025), 论文编号 e11}] 提出了关于刚性生成子图填充的两个尖锐连通性猜想：一个针对等维情形，另一个针对一个 $d$-刚性生成子图与一棵生成树的填充。我们证明了一个统一定理：对于任意正整数 $d_1,\ldots,d_s$，每个 $\sum_{i=1}^{s}d_i(d_i+1)$-连通图都包含两两边不相交的生成子图 $H_1,\ldots,H_s$，使得每个 $H_i$ 都是 $d_i$-刚性的。当 $\sum_{i=1}^{s}d_i(d_i+1)\ge4$ 时，该连通性界是尖锐的。作为特例，该定理解决了这两个猜想，确认了 Garamvölgyi、Jordán 和 Király [{\bf J. Combin. Theory Ser. B} \textbf{166} (2024), 1--29] 的猜想，即每个 $tk(k+1)$-连通图包含 $t$ 个两两边不相交的 $k$-连通生成子图，并给出了将一个 $d$-刚性生成子图与 $r$ 棵两两边不相交的生成树打包的尖锐阈值 $d(d+1)+2r$。我们还获得了与 Thomassen 关于图的高度连通定向的猜想相关的两个上界。如果 $f(q)$ 是使得每个 $f(q)$-连通图都有 $q$-连通定向的最小整数，那么对于每个 $q\ge3$，$f(q)\le(25q^2+41q-16)/2$，并且对于所有足够大的 $q$，$f(q)\le8q^2+212q+1404=(8+o(1))q^2$；这两个结果将先前二次界中的首项系数从 $320$ 分别降低到 $25/2$（对于每个 $q\ge3$）和 $8$（对于所有足够大的 $q$）。与 Garamvölgyi 等人获得的 $f(q)$ 的界相比，我们获得了更好的界，这不仅得益于我们紧的刚性结果，还利用了在移除两个边不相交的生成（充分）刚性图后剩余的边。
英文摘要
Garamvölgyi, Jordán, Király and Villányi [{{\bf Forum Math. Pi} \textbf{13} (2025), Paper No.~e11}] posed two sharp connectivity conjectures for packing rigid spanning subgraphs: one for the equal-dimensional case and the other for the packing of a $d$-rigid spanning subgraph with a spanning tree. We prove a unified theorem: for arbitrary positive integers $d_1,\ldots,d_s$, every $\sum_{i=1}^{s}d_i(d_i+1)$-connected graph contains pairwise edge-disjoint spanning subgraphs $H_1,\ldots,H_s$ such that $H_i$ is $d_i$-rigid for every $i$. The connectivity bound is sharp whenever $\sum_{i=1}^{s}d_i(d_i+1)\ge4$. As special cases, the theorem settles both conjectures, confirms the conjecture of Garamvölgyi, Jordán and Király [{\bf J. Combin. Theory Ser. B} \textbf{166} (2024), 1--29] that every $tk(k+1)$-connected graph contains $t$ pairwise edge-disjoint $k$-connected spanning subgraphs, and gives the sharp threshold $d(d+1)+2r$ for packing one $d$-rigid spanning subgraph together with $r$ pairwise edge-disjoint spanning trees. We also obtain two upper bounds related to Thomassen's conjecture on highly connected orientations of graphs. If $f(q)$ is the least integer such that every $f(q)$-connected graph has a $q$-connected orientation, then $f(q)\le(25q^2+41q-16)/2$ for every $q\ge3$ and $f(q)\le8q^2+212q+1404=(8+o(1))q^2$ for all sufficiently large $q$; these two results reduce the leading coefficient in the previous quadratic bound from $320$ to $25/2$ for every $q\ge3$ and $8$ for all sufficiently large $q$. Compared to the bound for $f(q)$ obtained by Garamvölgyi et al. we obtain the better bounds, not only through our tight rigidity result but also by exploiting the leftover edges when we remove two edge-disjoint spanning (sufficiently) rigid graphs.
Comments23 pages
