Sharp connectivity thresholds for mixed rigidity packings and improved bounds for highly connected orientations of graphs

Abstract

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.