Nonemptiness problems of Wang tiles with three colors

Hung-Hsun Chen National Chiao Tung University Wen-Guei Hu National Chiao Tung University De-Jan Lai National Chiao Tung University Song-Sun Lin National Chiao Tung University

Dynamical Systems mathscidoc:1609.11002

Theoretical Computer Science, 547, 34-45, 2014
This investigation studies nonemptiness problems of plane edge coloring with three colors. 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 nonemptiness problem is 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. Wang’s conjecture is that for any B of Wang tiles, Σ(B) 6= ∅ if and only if P(B) 6= ∅, where P(B) is the set of all periodic patterns on Z2 that can be generated by the tiles in B. When p ≥ 5, Wang’s conjecture is known to be wrong. When p = 2, theconjecture is true. This study proves that when p = 3, the conjecture is also true. 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(B ˜) 6= ∅ for B ˜ % B. Wang’s conjecture is shown to be true by proving that B ∈ N(3) implies Σ(B) = ∅.
Nonemptiness problem; Wang tile; Wang's conjecture; aperiodic set
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@inproceedings{hung-hsun2014nonemptiness,
  title={Nonemptiness problems of Wang tiles with three colors},
  author={Hung-Hsun Chen, Wen-Guei Hu, De-Jan Lai, and Song-Sun Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914155709798711004},
  booktitle={ Theoretical Computer Science},
  volume={547},
  pages={34-45},
  year={2014},
}
Hung-Hsun Chen, Wen-Guei Hu, De-Jan Lai, and Song-Sun Lin. Nonemptiness problems of Wang tiles with three colors. 2014. Vol. 547. In Theoretical Computer Science. pp.34-45. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914155709798711004.
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