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

Abstract

Let $$k\ge 2, p\ge 1, q\ge 0$$kź2,pź1,qź0 be integers. We prove that every $$(4kp-2p+2q)$$(4kp-2p+2q)-connected graph contains p spanning subgraphs $$G_i$$Gi for $$1\le i\le p$$1≤i≤p and q spanning trees such that all $$p+q$$p+q subgraphs are pairwise edge-disjoint and such that each $$G_i$$Gi is k-edge-connected, essentially $$(2k-1)$$(2k-1)-edge-connected, and $$G_i -v$$Gi-v is $$(k-1)$$(k-1)-edge-connected for all $$v\in V(G)$$vźV(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 $$(6p+2q)$$(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.