Dirac's Condition for Completely Independent Spanning Trees

Two spanning trees T 1 and T 2 of a graph G are completely independent if, for any two vertices u and v , the paths from u to v in T 1 and T 2 are internally disjoint. In this article, we show two sufficient conditions for the existence of completely independent spanning trees. First, we show that a graph of n vertices has two completely independent spanning trees if the minimum degree of the graph is at least . Then, we prove that the square of a 2‐connected graph has two completely independent spanning trees. These conditions are known to be sufficient conditions for Hamiltonian graphs.