Approximate Itai–Zehavi conjecture for random graphs

A famous conjecture by Itai and Zehavi states that, for every 𝑑 -vertex-connected graph 𝐺 and every vertex 𝑟 in 𝐺 , there are 𝑑 spanning trees of 𝐺 such that, for every vertex 𝑣 in 𝐺 ∖{𝑟} , the paths between 𝑟 and 𝑣 in different trees are internally vertex-disjoint. We show that with high pro