| Abstract: |
| 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 $\\Sigma(B) \\neq \\empty$ , where $\\Sigma(B)$ is the set of all global patterns on $\\mathbb{Z}^2$ that can be constructed from the Wang tiles in $B$. Wang`s conjecture is that for any $B$ of Wang tiles, $\\Sigma(B)\\neq \\empty$ if and only if $P(B)\\neq \\empty$, where $P(B)$ is the set of all periodic patterns on $\\mathbb{Z}^2$ that can be generated by the tiles in $B$.
When $p \\geq 5$, Wang`s conjecture is known to be wrong. When $p = 2$, the conjecture is true. This study proves that when $p = 3$, the conjecture is also true. If $P(B) \\neq \\empty$, then $B$ has a subset $B^`$ of minimal cycle generators such that $P(B^` )\\neq \\empty$ and $P(B^{``} )= \\empyt$ for $B^{``} \\subsetneqq B^`$. 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 \\in N(3)$, then $ P(B)= \\empyt$ and $P(\\bar{B})\\neq \\empty for $\\bar{B} \\supsetneqq B$. Wang`s conjecture is shown to be true by proving that $B \\in N(3)$ implies $\\Sigma (B) = \\empty$. |
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