The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these notions by defining (i, j)-disjoint spanning trees, where i (j, respectively) is the number of vertices (edges, respectively) that are shared by more than one tree. We illustrate how (i, j)-disjoint spanning trees provide some nuances between the existence of disjoint connected dominating sets and completely independent spanning trees. We prove that determining if there exist two (i, j)-disjoint spanning trees in a graph G is NP-complete, for every two positive integers i and j. Moreover we prove that for square of graphs, k-connected interval graphs, complete graphs and several grids, there exist (i, j)-disjoint spanning trees for interesting values of i and j.
 Let T1, T2, …, Tk be spanning trees in a graph G. If for any two vertices u, v in G, the paths from u to v in T1, T2, …, Tk are pairwise internally disjoint, then T1, T2, …, Tk are completely independent spanning trees in G. Completely independent spanning trees can be applied to fault-tolerant communication problems in interconnection networks. In this article, we show that there are two completely independent spanning trees in any torus network. Besides, we generalize the result for the Cartesian product. In particular, we show that there are two completely independent spanning trees in the Cartesian product of any 2-connected graphs. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012 © 2012 Wiley Periodicals, Inc.   Let G be a graph. Let T1, T2, . . . , Tk be spanning trees in G. If for any two vertices u, v in G, the paths from u to v in T1, T2, . . . , Tk are pairwise openly disjoint, then we say that T1, T2, . . . , Tk are completely independent spanning trees in G. In this paper, we show that there are two completely independent spanning trees in any 4-connected maximal planar graph. Our proof induces a linear-time algorithm for finding such trees. Besides, we show that given a graph G, the problem of deciding whether there exist two completely independent spanning trees in G is NP-complete. Abstract Two edge-disjoint spanning trees of a graph G are completely independent if the two paths connecting any two vertices of G in the two trees are internally disjoint. It has been asked whether sufficient conditions for hamiltonian graphs are also sufficient for the existence of two completely independent spanning trees (CISTs). We prove that it is true for the classical Ore-condition. That is, if G is a graph on n vertices in which each pair of non-adjacent vertices have degree-sum at least n , then G has two CISTs. It is known that the line graph of every 4-edge connected graph is hamiltonian. We prove that this is also true for CISTs: the line graph of every 4-edge connected graph has two CISTs. Thomassen conjectured that every 4-connected line graph is hamiltonian. Unfortunately, being 4-connected is not enough for the existence of two CISTs in line graphs. We prove that there are infinitely many 4-connected line graphs that do not have two CISTs.   Two spanning trees T1 and T2 of a graph G are completely independent if, for any two vertices u and v, the paths from u to v in T1 and T2 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.   Completely independent spanning trees (T_1,T_2, ,T_k ) in a graph G are spanning trees in G such that for any pair of distinct vertices u and v, the k paths in the spanning trees between u and v mutually have no common edge and no common vertex except for u and v. The concept finds applications in fault-tolerant communication problems in a network. Recently, it was shown that Dirac’s condition for a graph to be hamiltonian is also a sufficient condition for a graph to have two completely independent spanning trees. In this paper, we generalize this result to three or more completely independent spanning trees. Namely, we show that for any graph G with (n 7 ) vertices, if the minimum degree of a vertex in G is at least (n-k ), where (3 k n 2 ), then there are ( n k ) completely independent spanning trees in G. Besides, we improve the lower bound of ( n 2 ) on the Dirac’s condition for completely independent spanning trees to ( n-1 2 ) except for some specific graph. Our results are theoretical ones, since these minimum degree conditions can be applied only to a very dense graph. We then present constructions of symmetric regular graphs which include optimal graphs with respect to the number of completely independent spanning trees. 