Problems on the Triangular Lattice
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917671968178176 |
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| author | Brouwer, Gaston A. Joe, Jonathan Noble, Abby A. Noble, Matt |
| author_facet | Brouwer, Gaston A. Joe, Jonathan Noble, Abby A. Noble, Matt |
| contents | In this work, we consider a number of problems defined on the triangular lattice with $n$ rows, which we will denote as $T_n$. Define a \textit{proper coloring} to be an assignment of colors to the points of $T_n$ such that no three points constituting the vertices of an equilateral triangle all receive the same color, and denote by $f(n)$ the smallest possible number of colors that can be used in a proper coloring of $T_n$. We either determine exactly or give upper bounds for $f(n)$ for many small values of $n$, and it is shown that $\lim_{n\to\infty} \frac{f(n)}{n} \leq \frac13$. We also give formulas counting the number of pairs of points in $T_n$ for which there are, respectively, 0, 1, or 2 choices of points in $T_n$ which extend those two into the vertices of an equilateral triangle. Along the way, we pose a number of related questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Problems on the Triangular Lattice Brouwer, Gaston A. Joe, Jonathan Noble, Abby A. Noble, Matt Combinatorics In this work, we consider a number of problems defined on the triangular lattice with $n$ rows, which we will denote as $T_n$. Define a \textit{proper coloring} to be an assignment of colors to the points of $T_n$ such that no three points constituting the vertices of an equilateral triangle all receive the same color, and denote by $f(n)$ the smallest possible number of colors that can be used in a proper coloring of $T_n$. We either determine exactly or give upper bounds for $f(n)$ for many small values of $n$, and it is shown that $\lim_{n\to\infty} \frac{f(n)}{n} \leq \frac13$. We also give formulas counting the number of pairs of points in $T_n$ for which there are, respectively, 0, 1, or 2 choices of points in $T_n$ which extend those two into the vertices of an equilateral triangle. Along the way, we pose a number of related questions. |
| title | Problems on the Triangular Lattice |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.12321 |