Extremal Topological and Geometric Problems for Polyominoes
Extremal Topological and Geometric Problems for Polyominoes
复制标题
多联骨牌的极值拓扑和几何问题
DOI：
--
复制
发表时间：
2020
影响因子：
0.7
通讯作者：
Erika Berenice Roldan
中科院分区：
文献类型：
--
作者：
Greg Malen;Erika Berenice Roldan
我们给出了一个完整的解决方案的极值拓扑组合问题的最小数量的瓷砖需要构建一个polyomino与$h$孔。我们表示这个数字的$g（h）$和我们分析的结构特性的polyominoes与$h$孔和$g（h）$瓷砖，其效率特征的拓扑等周不等式，涉及最小周长，面积的孔，和结构的对偶图的polyomino。对于$hleqslane 8$，$g（h）$的值最初是由Tomas Olivera e Silva在2015年计算的，而对于序列$h_l=（2^{2l}-1）/3$，Kahle和Róldan-Roa在2019年也证明了$g（h）近似于2 h $。在这里我们还证明了Kahle和Róldan-Roa构造的具有$h_l=（2^{2l}-1）/3$洞和$g（h_l）$瓦片的多项式序列，在达到这些极值拓扑性质方面，实际上是唯一的，直到等距;也就是说，对于$h_l$洞具有最小瓦片数。
We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and we analyze structural properties of polyominoes with $h$ holes and $g(h)$ tiles, characterizing their efficiency by a topological isoperimetric inequality that relates minimum perimeter, the area of the holes, and the structure of the dual graph of a polyomino. For $hleqslant 8$ the values of $g(h)$ were originally computed by Tomas Olivera e Silva in 2015, and for the sequence $h_l=(2^{2l}-1)/3$ by Kahle and Róldan-Roa in 2019, who also showed that asymptotically $g(h) approx 2h$. Here we also prove that the sequence of polyominoes constructed by Kahle and Róldan-Roa that have $h_l=(2^{2l}-1)/3$ holes and $g(h_l)$ tiles, are in fact unique up to isometry with respect to attaining these extremal topological properties; that is, having the minimal number of tiles for $h_l$ holes.