Counterexamples to the uniform shortest path routing conjecture for vertex-transitive graphs

In this note we disprove the uniform shortest path routing conjec ture for vertex transitive graphs An in nite family of counterexam ples is given Let G be a connected graph on a vertex set V A routing in G is a set R fPuvj u v V V u vg of jV j jV j paths in G where each individual path Puv has initial vertex u and terminal vertex v We note that the paths Puv and Pvu may be di erent We say that R is a shortest path routing if the length of each path Puv R is equal to the distance d u v of the vertices u and v in the graph G Any subset R of a routing R is a partial routing For any vertex w of G the load of w in a partial routing R R denoted by R w is the number of paths in R that pass through w i e that contain w as an internal vertex For a routing R let R be the maximum of the loads R w over all vertices w of G The vertex forwarding index G of the graph G is the minimum of R over all routings R in G Routings and the associated forwarding indices have been studied exten sively see e g for a most recent survey we recommend from which the following basic facts can be extracted Proposition Let G be a connected graph of order n Then