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 Σ(B) 6= ∅, where Σ(B) is the set of all global patterns on Z2 that can be constructed from the Wang tiles in B. When p ≥ 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. P(B) is the set of all periodic patterns on Z2 that can be generated by the tiles in B. If P(B) 6= ∅, then B has a subset B′ of minimal cycle generators such that P(B′) 6= ∅ and P(B′′) = ∅ for B′′ $ B′. This study demonstrates that the set C(3) of all minimal cycle generators contains 787, 605 members that can be classified into 2, 906 equivalence classes. N (3) is the set of all maximal non-cycle generators : if B ∈ N (3), then P(B) = ∅ and P(B̃) 6= ∅ for B̃ % B. The problem is shown to be decidable by proving that B ∈ N (3) implies Σ(B) = ∅. Consequently, Σ(B) 6= ∅ if and only if P(B) 6= ∅.