# On the maximum degree of path-pairable planar graphs

## TL;DR

This paper proves that in any path-pairable planar graph with n vertices, there must be a vertex with degree proportional to n, highlighting a fundamental property of such graphs.

## Contribution

It establishes a lower bound on the maximum degree in path-pairable planar graphs, revealing a key structural limitation.

## Key findings

- Any n-vertex path-pairable planar graph has a vertex with degree linear in n.
- The result constrains the possible degree distributions in such 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 investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any $n$-vertex path-pairable planar graph must contain a vertex of degree linear in $n$.