On Independent Spanning Trees in Random Graphs

A central challenge in network design is ensuring resilience: how can we guarantee multiple, independent, communication pathways between nodes, even when some connections fail in a network? In 1989, Zehavi and Itai formulated a graph-theoretic conjecture that captures the essence of this problem. They proposed that any k-vertex-connected graph contains k independent spanning trees rooted at any given root r, which means that for every vertex v in the graph, the unique r–v paths within these k spanning trees are entirely disjoint, apart from their endpoints r and v. Despite decades of effort, this conjecture has only been proven for k ≤ 4 and for specific graph families using their underlying topological structure, leaving the general case as an open problem in graph theory with substantial consequences in the field of distributed algorithms. We make significant progress on the Zehavi-Itai conjecture by proving it holds for almost all graphs of relevant densities. More precisely, we show that there exists some constant C > 1 such that for all C log n/n ≤ p < 0.99, the binomial random graph G(n, p) contains a family of δ(G) independent spanning trees rooted at any given vertex r with high probability. Note that the lower bound on p up to the constant C matches the standard threshold for connectivity for G(n, p), thus we establish an essentially best possible result for random graphs.