Escaping from a quadrant of the $6\times 6$ grid by edge disjoint paths

TL;DR

This paper proves that in a 6x6 grid, four pairs of terminals can be connected by edge-disjoint paths, even when some terminals are confined within a quadrant, by establishing key lemmas for such escape scenarios.

Contribution

The paper introduces new lemmas that enable connecting terminal pairs in a grid despite quadrant confinement, advancing understanding of edge-disjoint path linkages in grid graphs.

Findings

Edge-disjoint paths can connect four terminal pairs in a 6x6 grid.

Terminals confined in a quadrant can be linked through specific escape lemmas.

The lemmas are proven for cases with up to four terminals in a quadrant.

Abstract

Let $G$ be the Cartesian product of two finite paths, called a grid, and let $T$ be the set of eight distinct vertices of $G$, called terminals. Assume that $T$ is partitioned into four terminal pairs $\{s_i,t_i\}$, $1\leq i\leq 4$, to be linked in $G$ by using edge disjoint paths. To prove that such a linkage always exists we need a sequence of technical lemmas making possible for some terminals to `escape' from a $3\times 3$ corner of $Q\subset G$, called a `quadrant'. Here we state those lemmas, and give a proof for the cases when $Q$ contains at most $4$ terminals.