A full characterization of invariant embeddability of unimodular planar graphs

IF 0.9 3区数学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Random Structures & Algorithms Pub Date : 2023-10-04 DOI:10.1002/rsa.21188

Ádám Timár, László Márton Tóth

引用次数: 2

Abstract

Abstract When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry‐invariant? This question was answered for one‐ended unimodular graphs in Benjamini and Timar, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends the question was left open. We revisit Halin's graph theoretic characterization of graphs that have a locally finite embedding into the plane. Then we prove that such unimodular random graphs do have a locally finite invariant embedding into the Euclidean or the hyperbolic plane, depending on whether the graph is amenable or not.

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单模平面图不变嵌入性的完整刻画

什么时候可以在欧几里得平面或双曲平面上绘制单模随机平面图，并且随机图的分布是等距不变的?这个问题在Benjamini和Timar的单端单模图中得到了解答，使用这样的图在平面上自动具有局部有限(单连通)图的事实。对于有多个端点的图，这个问题还没有解决。我们重新审视Halin的图论特征图有一个局部有限嵌入到平面。然后我们证明了这类单模随机图在欧几里德平面或双曲平面上确实有局部有限不变量嵌入，这取决于图是否可服从。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Random Structures & Algorithms 数学-计算机：软件工程

CiteScore

2.50

自引率

10.00%

发文量

审稿时长

>12 weeks

期刊介绍： It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness. Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.