论文标题
无限平面图的尖锐等级不等式，有界顶点和面部度
Sharp isoperimetric inequalities for infinite plane graphs with bounded vertex and face degrees
论文作者
论文摘要
我们为具有边界顶点和面部度的无限平面图（Tessellations）的等速标准提供了尖锐的边界。例如，如果$ g $是满足不平等的平面图，$ p_1 \ leq \ mbox {deg} \ v \ v \ leq p_2 $ for $ v \ in v（g）$ in v（g）$和$ q_1 \ leq \ leq \ mbox {deg} $ q_2 $是自然数，因此$ 1/p_i + 1/q_i \ leq 1/2 $，$ i = 1,2 $，然后我们表明\ [φ（p_1，q_1）\ leq \ leq \ leq \ inf_s \ frac {| \ frac {| \ partial s |}有限的非空子图$ s \ subset g $，$ \ partial s $是将$ s $连接到$ g \ setminus s $的边缘集，而$φ（p，q）$由\ [φ（p，q）=（p，q）=（p-2）\ sqrt \ sqrt {1- \ freac {1- \ freac {p，p，q）定义。 \ \]对于$ p_1 = 3 $，这给出了2002年Lawrencenko，Plummer和Zha的猜想的肯定答案，对于一般$ P_I $和$ Q_I $，我们的结果完全解决了Lyons和Peres在2016年的书中提出的问题，从而扩展了Lawrencenko等人的猜测。到上述形式。
We provide sharp bounds for the isoperimetric constants of infinite plane graphs (tessellations) with bounded vertex and face degrees. For example, if $G$ is a plane graph satisfying the inequalities $p_1 \leq \mbox{deg}\ v \leq p_2$ for $v \in V(G)$ and $q_1 \leq \mbox{deg}\ f \leq q_2$ for $f \in F(G)$, where $p_1, p_2, q_1$, and $q_2$ are natural numbers such that $1/p_i + 1/q_i \leq 1/2$, $i=1,2$, then we show that \[ Φ(p_1, q_1) \leq \inf_S \frac{|\partial S|}{|V(S)|} \leq Φ(p_2, q_2), \] where the infimum is taken over all finite nonempty subgraphs $S \subset G$, $\partial S$ is the set of edges connecting $S$ to $G \setminus S$, and $Φ(p,q)$ is defined by \[ Φ(p, q) = (p-2) \sqrt{1 - \frac{4}{(p-2)(q-2)}}. \] For $p_1=3$ this gives an affirmative answer to a conjecture by Lawrencenko, Plummer, and Zha from 2002, and for general $p_i$ and $q_i$ our result fully resolves a question posed in the book by Lyons and Peres from 2016, where they extended the conjecture of Lawrencenko et al. to the above form.
