用三種顏色點著色的非空問題

Abstract

這個研究探討用三種顏色來著點的非空性問題。 點著色中，方格中每個點從三種顏色中找一種顏色來著色， 相接的點是一樣的顏色就可以接起來。 非空問題是給定一組tile 後，是否可以用這組tile 去拼出 整個平面。而王浩的猜測是說給定一組tile，如果可以用這 組tile 拼出全平面，就存在週期性的拼法去拼出來。 但當p大於等於6時,王浩的猜測被推翻了，而在p=2 時被證實是對的。 P=3,4,5 是我們還不知道的，我這篇主要探討當p=3 時王浩 的猜測會不會是對的。

This investigation studies nonemptiness problems of plane corner coloring with three colors. In the corner coloring of a plane, unit squares with colored corners that have one of p colors are arranged side by side such that the touching corners of the adjacent tiles have the same colors. Given a basic set of tiles, the nonemptiness problem is to determine whether or not Σ(ℬ)≠⌀ , where Σ(ℬ) is the set of all global patterns on ℤ2 that can be constructed from the tiles in ℬ. Wang's conjecture is that for any ℬ of tiles, if and only if P(ℬ)≠⌀ , 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 in corner coloring is known to be wrong. When p = 2, the conjecture is true. Therefore, p=3,4,5 are the cases we havn’t known yet. We study when p=3 whether Wang's conjecture can be hold or not.