On the maximum diameter of path-pairable graphs

A graph is path-pairable if for any pairing of its vertices there exist edge disjoint paths joining the vertices in each pair. We obtain sharp bounds on the maximum possible diameter of path-pairable graphs which either have a given number of edges, or are c- degenerate. Along the way we show that a large family of graphs obtained by blowing up a path is path-pairable, which may be of independent interest.