Under what conditions can we construct 2 edge-disjoint trees in a finite undirected graph? I think might be addressed by "On the Problem of Decomposing a Graph into $n$ Connected Factors" by W. T. Tutte, but I can't find a free pdf of the paper online. If somebody can present the results for the case of 2 trees here, I'd really appreciate it!
2026-03-28 11:42:52.1774698172
2 edge-disjoint trees in graph
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This is a case of the Nash-Williams-Theorem: We use the following notation: If $(X_1, \dots, X_p)$ is a partition of $V(G)$ into $p$ parts, then $\delta(X_1, \dots, X_p)$ denotes the set of edges, which have endpoints in different parts of the partition. The Nash-Williams-Theorem states, that $G$ contains $k$ edge-disjoint spanning trees, if and only $|\delta(X_1, \dots, X_p)| \geq k(p-1)$ for all partitions $(X_1, \dots, X_p)$. You want the case $k=2$. There is a (short) Wikipedia article about the theorem and it also seems to be in the standard book by Diestel (haven't personally checked).