4-tangrams are 4-avoidable
A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one corresponds to squares. For $...
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| Published in: | Discrete Mathematics and Theoretical Computer Science Vol. 27:3; no. Combinatorics; pp. 1 - 6 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
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01.10.2025
Discrete Mathematics & Theoretical Computer Science |
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| ISSN: | 1365-8050, 1462-7264, 1365-8050 |
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| Abstract | A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one corresponds to squares. For $k\ge1$, let $t(k)$ denote the minimum size of an alphabet over which an infinite word avoids tangrams with cut number at most~$k$. The existence of infinite ternary square-free words shows that $t(1)=t(2)=3$. We show that $t(3)=t(4)=4$, answering a question from Dębski, Grytczuk, Pawlik, Przybyło, and Śleszyńska-Nowak. |
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| AbstractList | A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one corresponds to squares. For $k\ge1$, let $t(k)$ denote the minimum size of an alphabet over which an infinite word avoids tangrams with cut number at most~$k$. The existence of infinite ternary square-free words shows that $t(1)=t(2)=3$. We show that $t(3)=t(4)=4$, answering a question from Dębski, Grytczuk, Pawlik, Przybyło, and Śleszyńska-Nowak. A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The cut number of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one correspond to squares. For k [greater than or equal to] 1, let t(k) denote the minimum size of an alphabet over which an infinite word avoids tangrams with cut number at most k. The existence of infinite ternary square-free words shows that t(1) = t(2) = 3. We show that t(3) = t(4) = 4, answering a question from Debski, Grytczuk, Pawlik, Przybylo, and Sleszynska-Nowak. A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The $\textit{cut number}$ of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one corresponds to squares. For $k\ge1$, let $t(k)$ denote the minimum size of an alphabet over which an infinite word avoids tangrams with cut number at most~$k$. The existence of infinite ternary square-free words shows that $t(1)=t(2)=3$. We show that $t(3)=t(4)=4$, answering a question from Dębski, Grytczuk, Pawlik, Przybyło, and Śleszyńska-Nowak. A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The cut number of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one correspond to squares. For k [greater than or equal to] 1, let t(k) denote the minimum size of an alphabet over which an infinite word avoids tangrams with cut number at most k. The existence of infinite ternary square-free words shows that t(1) = t(2) = 3. We show that t(3) = t(4) = 4, answering a question from Debski, Grytczuk, Pawlik, Przybylo, and Sleszynska-Nowak. Keywords: combinatorics on words |
| Audience | Academic |
| Author | Ochem, Pascal Pierron, Théo |
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| DOI | 10.46298/dmtcs.15310 |
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| Snippet | A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The... A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The cut... |
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| Title | 4-tangrams are 4-avoidable |
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