Disjoint T-paths in tough graphs

Let G be a graph and T a set of vertices. A T -path in G is a path that begins and ends in T , and none of its internal vertices are contained in T . We define a T -path covering to be a union of vertex-disjoint T paths spanning all of T . Concentrating on graphs that are tough (the removal of any nonempty set X of vertices yields at most |X| components), we completely characterize the edges that are contained in some T -path covering. Our main tool is Mader’s S-paths theorem. A corollary of our result is that each edge of a k-regular k-edge-connected graph (k ≥ 2) is contained in a T -path covering. This is, in a sense, a best possible counterpart of the result of Plesńık that every edge of a k-regular (k− 1)edge-connected graph of even order is contained in a 1-factor.