Plane Triangulations(平面三角剖分)研究综述
Plane Triangulations 平面三角剖分 - As corollaries, we have shown that the conjecture holds for subdivisions of plane triangulations, simple $3$-connected plane quadrangulations, and simple $3$-connected plane pentagulations with an even maximum face degree, for regular subdivisions of simple $3$-connected plane graphs of maximum degree at least 10, and for subdivisions of simple $3$-connected plane graphs having the maximum face degree large enough in comparison with the number of vertices of their longest paths consisting only of vertices of degree two. [1] We prove that plane graphs of minimum degree at least $7$ have site percolation threshold bounded away from $1/2$, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini and Horesh that the critical probability is at most $1/2$ for plane triangulations of minimum degree $6$. [2] Denote by $M_t$ a matching of size $t$ and $\mathcal {T}_n$ the class of all plane triangulations of order $n$, respectively. [3] The set of all plane triangulations of order n is denoted by T n. [4] This bound is improved to six for several wide families of plane graphs and to four for plane triangulations. [5] So far, such a tight description has been obtained only for several restricted classes of 3-polytopes: those with minimum degree 5 (Borodin, 1989), without vertices of degree 3 (Borodin and Ivanova, 2013), for plane triangulations (Borodin et al. [6] Motivated by anti-Ramsey numbers introduced by Erd\H{o}s, Simonovits and S\'os in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. [7] Dujmovi c, Eppstein, Suderman, and Wood showed that every 3-connected plane graph G with n vertices admits a straight-line drawing with at most 2:5n 3 segments, which is also the best known upper bound when restricted to plane triangulations. [8] Let T n denote the class of all plane triangulations of order n. [9] Given a positive integer $n$ and a planar graph $H$, let $\mathcal{T}_n(H)$ be the family of all plane triangulations $T$ on $n$ vertices such that $T$ contains a subgraph isomorphic to $H$. [10] Denote by $kK_2$ a matching of size $k$ and $\mathcal {T}_n$ the class of all plane triangulations of order $n$, respectively. [11]作为推论，我们已经证明该猜想适用于平面三角剖分、简单的$3$-连通平面四边形和具有最大面度的简单$3$-连通平面五边形，对于简单$3$-连通平面图的常规细分最大度数至少为 10，并且对于简单的 $3$ 连通平面图的细分，其最大面度数与其最长路径的仅由二度数的顶点组成的顶点数相比足够大。 [1] 我们证明了 Benjamini 和 Schramm 猜想的最小度数至少为 $7$ 的平面图具有远离 $1/2$ 的站点渗透阈值，并且在 Angel、Benjamini 和 Horesh 的一个猜想上取得了进展，即临界概率为对于最小度数为 6 美元的平面三角测量，最多 1/2 美元。 [2] 用 $M_t$ 分别表示大小为 $t$ 和 $\mathcal {T}_n$ 的所有平面三角剖分的类 $n$。 [3] 所有 n 阶平面三角剖分的集合用 T n 表示。 [4] 对于几个广泛的平面图族，此界限改进为六个，而平面三角剖分则为四个。 [5] 到目前为止，仅对几个受限类别的 3 多面体获得了如此严格的描述：最小度数为 5 的那些（Borodin，1989），没有度数为 3 的顶点（Borodin 和 Ivanova，2013），平面三角剖分（Borodin 等人。 [6] 受 Erd\H{o}s、Simonovits 和 S\'os 在 1975 年引入的反拉姆齐数的启发，我们研究了宿主图是平面三角剖分时的反拉姆齐问题。 [7] Dujmovi c、Eppstein、Suderman 和 Wood 表明，每个具有 n 个顶点的 3 连通平面图 G 都允许有最多 2:5n 3 段的直线绘图，这也是平面三角剖分时最著名的上界。 [8] 令 T n 表示所有 n 阶平面三角剖分的类。 [9] 给定一个正整数 $n$ 和一个平面图 $H$，令 $\mathcal{T}_n(H)$ 是在 $n$ 个顶点上的所有平面三角剖分 $T$ 的族，使得 $T$ 包含一个子图与 $H$ 同构。 [10] 分别用 $kK_2$ 表示大小为 $k$ 和 $\mathcal {T}_n$ 的所有平面三角剖分的类的大小为 $n$ 的匹配。 [11]
Connected Plane Triangulations
We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most $\sqrt{2n}$ lines each of them horizontal or vertical. [1] Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. [2] As a direct consequence, we obtain that for $4$-connected plane triangulations there is a morph between every pair of RT-representations where the ``top-most'' triangle in both representations corresponds to the same vertex. [3]我们展示了可以重新绘制 4 连通平面三角剖分，使得边由直线段表示，顶点由一组最多 $\sqrt{2n}$ 条线覆盖，每条线都是水平或垂直的。 [1] 横向结构（也称为规则边缘标记）是在具有四边形外表面的 4 连接平面三角剖分上定义的组合结构。 [2] 作为一个直接的结果，我们得到对于 $4$ 连接的平面三角剖分，每对 RT 表示之间都有一个变形，其中两个表示中的“最顶部”三角形对应于相同的顶点。 [3]