顶点和面度数有界的无限平面图的精确等周不等式
Sharp isoperimetric inequalities for infinite plane graphs with bounded vertex and face degrees
AI总结：
本研究推导了顶点和面度数有界的无限平面图等周常数的精确界，验证了相关猜想并解决了推广后的公开问题。
摘要：
我们给出了顶点和面度数有界的无限平面图（镶嵌图）等周常数的精确界。例如，若平面图$G$满足：对$v \in V(G)$有$p_1 \leq \mbox{deg}\\\\ v \leq p_2$，对$f \in F(G)$有$q_1 \leq \mbox{deg}\\\\ f \leq q_2$，其中$p_1,p_2,q_1,q_2$为自然数，且满足$1/p_i + 1/q_i \leq 1/2$（$i=1,2$），则我们证明：\\[ Φ(p_1, q_1) \leq \inf_S \frac{|\partial S|}{|V(S)|} \leq Φ(p_2, q_2), \\] 其中下确界取遍$G$中所有有限非空子图$S$，$\partial S$是连接$S$与$G \setminus S$的边集，$Φ(p,q)$定义为\\[ Φ(p, q) = (p-2) \sqrt{1 - \frac{4}{(p-2)(q-2)}}. \\] 当$p_1=3$时，该结果对Lawrencenko、Plummer和Zha在2002年提出的一个猜想给出了肯定回答；对一般的$p_i$和$q_i$，我们的结果完全解决了Lyons和Peres在2016年的著作中提出的问题——他们将Lawrencenko等人的猜想推广为上述形式。
英文摘要：
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.
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