It is well known that any spanning tree of a grdph can be obtained from any other by a sequence of single edge exchanges in a way that preserves, at each step, the property of being a spanning tree We consider a variation of this problem concerning pairs of edge-disjoint spanning trees In particular, it is shown that any pair of edge-disjoint spanning trees can be obtained from any other by a sequence of single edge exchanges in a way that preserves, at each step, the property of being edge-disjoint spanning trees

Let G be a connected graph with edge set E embedded in the surface C. Let G" denote the geometric dual of G. For a subset d of E, let d denote the edges of G" that are dual to those edges of G in d. We prove the following generalizations of well-known facts about graphs embedded in the plane.(1) b is a boundary cycle in G if and only if 76 is a cocycle in G".(2) If T is a spanning tree of G, then T(E\T) contains a spanning tree of Go.(3) Let T be any spanning tree of G and, for e E E\T, let T(e) denote the fundamental cycle of e. Let U C E\T. Then TU is a spanning tree of Go if and only if the set of face boundaries, less any one, together with the set {T(e); e E E\(T U U)} is a basis for the cycle space of G.

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