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On 3*-connected graphs

Menger's Theorem states that in a 3-connected graph, any two vertices are joined by three openly disjoint paths. Here we consider 3-connected cubic graphs where two vertices exist so that the three disjoint paths between them contain all of the vertices of the graph (we call these graphs 3*connected); and also where the latter is true for ALL pairs of vertices (globally 3*-connected). A necessary condition for 3*-connectedness is that the circumference of the graph be at least 2(n + 1)/3 where n is the size of the vertex set. Global 3*-connectedness is not as strong as it might first appear. Many graphs have this property. In particular the generalized Petersen graphs, P(n, 2), are globally 3*-connected if and only if n 1, 3 (mod 6). The exceptions here are 3* -connected.