Note on Terminal-Pairability in Complete Grid Graphs

TL;DR

This paper proves that high-dimensional complete grid graphs are path-pairable, generalizing previous results, and improves the bounds on the maximum degree needed for path-pairability in such graphs.

Contribution

It confirms and extends the path-pairability of complete grid graphs and refines the maximum degree estimates for path-pairable graphs.

Findings

Complete grid graphs are path-pairable in high dimensions.

Improved bounds on maximum degree for path-pairability.

Generalization of previous results on path-pairability.

Abstract

We affirmatively answer and generalize the question of Kubicka, Kubicki and Lehel concerning the path-pairability of high-dimensional complete grid graphs. As an intriguing by-product of our result we significantly improve the estimate of the necessary maximum degree in path-pairable graphs, a question originally raised and studied by Faudree, Gy\'arf\'as, and Lehel.