Path transferability of graphs with bounded minimum degree

We consider a path as an ordered sequence of distinct vertices with a head and a tail. Given a path, a transfer-move is to remove the tail and add a vertex at the head. A graph is nn-transferable if any path with length nn can be transformed into any other such path by a sequence of transfer-moves. We show that, unless it is complete or a cycle, a connected graph is δδ-transferable, where δ≥2δ≥2 is the minimum degree.