Disjoint T-paths in tough graphs

Abstract

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 -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 Plesník that every edge of a k-regular (k-1)-edge-connected graph of even order is contained in a 1-factor. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 1–10, 2008