On the maximum diameter of path-pairable graphs

TL;DR

This paper establishes sharp bounds on the maximum diameter of path-pairable graphs with constraints on edges or degeneracy, and introduces a new family of path-pairable graphs via path blow-ups.

Contribution

It provides the first sharp bounds on diameter for constrained path-pairable graphs and identifies a new class of such graphs through path blow-ups.

Findings

Sharp bounds on maximum diameter for graphs with fixed edges or degeneracy.

Identification of a large family of path-pairable graphs via path blow-ups.

New insights into the structure of path-pairable graphs.

Abstract

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.