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RESEARCH PRODUCT

Almost disjoint spanning trees

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International audience; In this extended abstract, we only consider connected graphs. Let k ≥ 2 be an integer and T 1 ,. .. , T k be spanning trees in a graph G. A vertex is said to be an inner vertex in a tree T if it has degree at least 2 in T. We denote by I(T) the set of inner vertices of tree T. The spanning trees T 1 ,. .. , T k are completely independent spanning trees if any vertex from G is an inner vertex in at most one tree among T 1 ,. .. , T k and the trees T 1 ,. .. , T k are pairwise edge-disjoint. Completely independent spanning trees were introduced by Hasunuma [4] and then have been studied on different classes of graphs, such as underlying graphs of line graphs [4], maximal planar graphs [5], Cartesian product of two cycles [6] and k-trees [10]. Moreover, determining if there exist two completely independent spanning trees in a graph G is a NP-hard problem [5]. Recently, sufficient conditions inspired by the sufficient conditions for hamiltonicity have been determined in order to guarantee the existence of several completely independent spanning trees: Dirac's condition [1] and Ore's condition [2]. Moreover, Dirac's condition has been generalized to more than two trees [7].