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 n/2. 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. (C) 2013 Wiley Periodicals, Inc.