Decidability of plane edge coloring with three colors

Abstract

This investigation studies the decidability problem of plane edge coloring with three symbols. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges that have one of $p$ colors are arranged side by side such that the touching edges of the adjacent tiles have the same colors. Given a basic set $B$ of Wang tiles, the decision problem is to find an algorithm to determine whether or not $\Sigma(B)\neq\emptyset$, where $\Sigma(B)$ is the set of all global patterns on $\mathbb{Z}^{2}$ that can be constructed from the Wang tiles in $B$. When $p\geq 5$, the problem is known to be undecidable. When $p=2$, the problem is decidable. This study proves that when $p=3$, the problem is also decidable. $\mathcal{P}(B)$ is the set of all periodic patterns on $\mathbb{Z}^{2}$ that can be generated by the tiles in $B$. If $\mathcal{P}(B)\neq\emptyset$, then $B$ has a subset $B'$ of minimal cycle generators such that $\mathcal{P}(B')\neq\emptyset$ and $\mathcal{P}(B")=\emptyset$ for $B"\subsetneqq B'$. This study demonstrates that the set $\mathcal{C}(3)$ of all minimal cycle generators contains $787,605$ members that can be classified into $2,906$ equivalence classes. $\mathcal{N}(3)$ is the set of all maximal non-cycle generators: if $B\in \mathcal{N}(3)$, then $\mathcal{P}(B)=\emptyset$ and $\mathcal{P}(\tilde{B})\neq\emptyset$ for $\tilde{B}\supsetneqq B$. The problem is shown to be decidable by proving that $B\in \mathcal{N}(3)$ implies $\Sigma(B)=\emptyset$. Consequently, $\Sigma(B)\neq\emptyset$ if and only if $\mathcal{P}(B)\neq\emptyset$.