We prove an ear-decomposition theorem for 4-edge-connected graphs and use it to prove that for every 4-edge-connected graph G and every r ∈ V (G), there is a set of four spanning trees of G with the following property. For every vertex in G, the unique paths back to r in each tree are edge-disjoint. Our proof implies a polynomial-time algorithm for constructing the trees. Yu [7], the case k = 4 of the Independent Tree Conjecture was proven by Curran, Lee, and Yu across two papers [2, 3]. The Independent Tree Conjecture is open for nonplanar graphs with k > 4. In 1992, Khuller and Schieber [8] published a later-disproven argument that the Independent Tree Conjecture implies the Edge-Independent Tree Conjecture. Gopalan and Ramasubramanian [4] demonstrated that Khuller and Schieber's proof fails, but salvaged the technique, and proved the case k = 3 of the Edge-Independent Tree Conjecture by reducing it to the case k = 3 of the Independent Tree Conjecture. Schlipf and Schmidt [10] provided an alternate proof of the case k = 3 of the Edge-Independent Tree Conjecture, which does not rely on the Independent Tree Conjecture. The case k = 4 of the Edge-Independent Tree Conjecture is proven here, while the case k > 4 remains open.
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