Spanning rigid subgraph packing and sparse subgraph covering

Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have applications in graph theory. We discover a sufficient partition condition of packing spanning rigid subgraphs and spanning trees. As a corollary, we show that a simple graph $G$ contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees if $G$ is $(4k+2l)$-edge-connected, and $G-Z$ is essentially $(6k+2l - 2k|Z|)$-edge-connected for every $Z\subset V(G)$. Thus every $(4k+2l)$-connected and essentially $(6k+2l)$-connected graph $G$ contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees. Utilizing this, we show that every $6$-connected and essentially $8$-connected graph $G$ contains a spanning tree $T$ such that $G-E(T)$ is $2$-connected. These improve some previous results. Sparse subgraph covering problems are also studied.