Packing spanning trees and spanning 2-connected k-edge-connected essentially $(2k-1)$-edge-connected subgraphs

Abstract

Let $k\ge 2, p\ge 1, q\ge 0$ be integers. We prove that every $(4kp-2p+2q)$-connected graph contains p spanning subgraphs $G_i$ for $1\le i\le p$ and q spanning trees such that all $p+q$ subgraphs are pairwise edge-disjoint and such that each $G_i$ is k-edge-connected, essentially $(2k-1)$-edge-connected, and $G_i -v$ is $(k-1)$-edge-connected for all $v\in V(G)$. This extends the well-known result of Nash-Williams and Tutte on packing spanning trees, a theorem that every 6p-connected graph contains p pairwise edge-disjoint spanning 2-connected subgraphs, and a theorem that every $(6p+2q)$-connected graph contains p spanning 2-connected subgraphs and q spanning trees, which are all pairwise edge-disjoint. As an application, we improve a result on k-arc-connected orientations.