The $6\times 6$ grid is $4$-path-pairable

TL;DR

This paper proves that the 6x6 grid graph can connect any four pairs of terminals with edge-disjoint paths, demonstrating a specific path-pairability property in grid graphs.

Contribution

It establishes that the 6x6 grid is 4-path-pairable, a new result in the study of path-pairability in grid graphs.

Findings

The 6x6 grid is 4-path-pairable.

Any four terminal pairs can be connected with edge-disjoint paths.

The result extends understanding of path-pairability in grid structures.

Abstract

Let $G=P_6\Box P_6$ be the $6\times 6$ grid, the Cartesian product of two paths of six vertices. Let $T$ be the set of eight distinct vertices of $G$, called terminals, and assume that $T$ is partitioned into four terminal pairs $\{s_i,t_i\}$, $1\leq i\leq 4$. We prove that $G$ is $4$-path-pairable, that is, for every $T$ there exist in $G$ pairwise edge disjoint $s_i,t_i$-paths, $1\leq i\leq 4$.