Random graphs embeddable in order-dependent surfaces

Colin McDiarmid, Sophia Saller
{"title":"Random graphs embeddable in order-dependent surfaces","authors":"Colin McDiarmid, Sophia Saller","doi":"10.1002/rsa.21199","DOIUrl":null,"url":null,"abstract":"Given a ‘genus function’ <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0001\" display=\"inline\" location=\"graphic/rsa21199-math-0001.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>g</mi>\n<mo>=</mo>\n<mi>g</mi>\n<mo stretchy=\"false\">(</mo>\n<mi>n</mi>\n<mo stretchy=\"false\">)</mo>\n</mrow>\n$$ g=g(n) $$</annotation>\n</semantics></math>, we let <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0002\" display=\"inline\" location=\"graphic/rsa21199-math-0002.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msup>\n<mrow>\n<mi mathvariant=\"script\">E</mi>\n</mrow>\n<mrow>\n<mi>g</mi>\n</mrow>\n</msup>\n</mrow>\n$$ {\\mathcal{E}}^g $$</annotation>\n</semantics></math> be the class of all graphs <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0003\" display=\"inline\" location=\"graphic/rsa21199-math-0003.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>G</mi>\n</mrow>\n$$ G $$</annotation>\n</semantics></math> such that if <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0004\" display=\"inline\" location=\"graphic/rsa21199-math-0004.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>G</mi>\n</mrow>\n$$ G $$</annotation>\n</semantics></math> has order <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0005\" display=\"inline\" location=\"graphic/rsa21199-math-0005.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>n</mi>\n</mrow>\n$$ n $$</annotation>\n</semantics></math> (i.e., has <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0006\" display=\"inline\" location=\"graphic/rsa21199-math-0006.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>n</mi>\n</mrow>\n$$ n $$</annotation>\n</semantics></math> vertices) then it is embeddable in a surface of Euler genus at most <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0007\" display=\"inline\" location=\"graphic/rsa21199-math-0007.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>g</mi>\n<mo stretchy=\"false\">(</mo>\n<mi>n</mi>\n<mo stretchy=\"false\">)</mo>\n</mrow>\n$$ g(n) $$</annotation>\n</semantics></math>. Let the random graph <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0008\" display=\"inline\" location=\"graphic/rsa21199-math-0008.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mi>R</mi>\n</mrow>\n<mrow>\n<mi>n</mi>\n</mrow>\n</msub>\n</mrow>\n$$ {R}_n $$</annotation>\n</semantics></math> be sampled uniformly from the graphs in <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0009\" display=\"inline\" location=\"graphic/rsa21199-math-0009.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msup>\n<mrow>\n<mi mathvariant=\"script\">E</mi>\n</mrow>\n<mrow>\n<mi>g</mi>\n</mrow>\n</msup>\n</mrow>\n$$ {\\mathcal{E}}^g $$</annotation>\n</semantics></math> on vertex set <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0010\" display=\"inline\" location=\"graphic/rsa21199-math-0010.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mo stretchy=\"false\">[</mo>\n<mi>n</mi>\n<mo stretchy=\"false\">]</mo>\n<mo>=</mo>\n<mo stretchy=\"false\">{</mo>\n<mn>1</mn>\n<mo>,</mo>\n<mi>…</mi>\n<mo>,</mo>\n<mi>n</mi>\n<mo stretchy=\"false\">}</mo>\n</mrow>\n$$ \\left[n\\right]=\\left\\{1,\\dots, n\\right\\} $$</annotation>\n</semantics></math>. Observe that if <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0011\" display=\"inline\" location=\"graphic/rsa21199-math-0011.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>g</mi>\n<mo stretchy=\"false\">(</mo>\n<mi>n</mi>\n<mo stretchy=\"false\">)</mo>\n</mrow>\n$$ g(n) $$</annotation>\n</semantics></math> is 0 then <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0012\" display=\"inline\" location=\"graphic/rsa21199-math-0012.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mi>R</mi>\n</mrow>\n<mrow>\n<mi>n</mi>\n</mrow>\n</msub>\n</mrow>\n$$ {R}_n $$</annotation>\n</semantics></math> is a random planar graph, and if <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0013\" display=\"inline\" location=\"graphic/rsa21199-math-0013.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>g</mi>\n<mo stretchy=\"false\">(</mo>\n<mi>n</mi>\n<mo stretchy=\"false\">)</mo>\n</mrow>\n$$ g(n) $$</annotation>\n</semantics></math> is sufficiently large then <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0014\" display=\"inline\" location=\"graphic/rsa21199-math-0014.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mi>R</mi>\n</mrow>\n<mrow>\n<mi>n</mi>\n</mrow>\n</msub>\n</mrow>\n$$ {R}_n $$</annotation>\n</semantics></math> is a binomial random graph <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0015\" display=\"inline\" location=\"graphic/rsa21199-math-0015.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>G</mi>\n<mo stretchy=\"false\">(</mo>\n<mi>n</mi>\n<mo>,</mo>\n<mfrac>\n<mrow>\n<mn>1</mn>\n</mrow>\n<mrow>\n<mn>2</mn>\n</mrow>\n</mfrac>\n<mo stretchy=\"false\">)</mo>\n</mrow>\n$$ G\\left(n,\\frac{1}{2}\\right) $$</annotation>\n</semantics></math>. We investigate typical properties of <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0016\" display=\"inline\" location=\"graphic/rsa21199-math-0016.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mi>R</mi>\n</mrow>\n<mrow>\n<mi>n</mi>\n</mrow>\n</msub>\n</mrow>\n$$ {R}_n $$</annotation>\n</semantics></math>. We find that for <i>every</i> genus function <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0017\" display=\"inline\" location=\"graphic/rsa21199-math-0017.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>g</mi>\n</mrow>\n$$ g $$</annotation>\n</semantics></math>, with high probability at most one component of <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0018\" display=\"inline\" location=\"graphic/rsa21199-math-0018.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mi>R</mi>\n</mrow>\n<mrow>\n<mi>n</mi>\n</mrow>\n</msub>\n</mrow>\n$$ {R}_n $$</annotation>\n</semantics></math> is non-planar. In contrast, we find a transition for example for connectivity: if <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0019\" display=\"inline\" location=\"graphic/rsa21199-math-0019.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>g</mi>\n<mo stretchy=\"false\">(</mo>\n<mi>n</mi>\n<mo stretchy=\"false\">)</mo>\n</mrow>\n$$ g(n) $$</annotation>\n</semantics></math> is <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0020\" display=\"inline\" location=\"graphic/rsa21199-math-0020.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>O</mi>\n<mo stretchy=\"false\">(</mo>\n<mi>n</mi>\n<mo stretchy=\"false\">/</mo>\n<mi>log</mi>\n<mi>n</mi>\n<mo stretchy=\"false\">)</mo>\n</mrow>\n$$ O\\left(n/\\log n\\right) $$</annotation>\n</semantics></math> and <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0021\" display=\"inline\" location=\"graphic/rsa21199-math-0021.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>g</mi>\n</mrow>\n$$ g $$</annotation>\n</semantics></math> is non-decreasing then <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0022\" display=\"inline\" location=\"graphic/rsa21199-math-0022.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mtext>lim inf</mtext>\n</mrow>\n<mrow>\n<mi>n</mi>\n<mo>→</mo>\n<mi>∞</mi>\n</mrow>\n</msub>\n<mi>ℙ</mi>\n<mo stretchy=\"false\">(</mo>\n<msub>\n<mrow>\n<mi>R</mi>\n</mrow>\n<mrow>\n<mi>n</mi>\n</mrow>\n</msub>\n<mspace width=\"0.3em\"></mspace>\n<mtext>is connected</mtext>\n<mo stretchy=\"false\">)</mo>\n<mo>&lt;</mo>\n<mn>1</mn>\n</mrow>\n$$ \\lim\\ {\\operatorname{inf}}_{n\\to \\infty}\\mathbb{P}\\left({R}_n\\kern0.3em \\mathrm{is}\\ \\mathrm{connected}\\right)&lt;1 $$</annotation>\n</semantics></math>, and if <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0023\" display=\"inline\" location=\"graphic/rsa21199-math-0023.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<mi>g</mi>\n<mo stretchy=\"false\">(</mo>\n<mi>n</mi>\n<mo stretchy=\"false\">)</mo>\n<mo>≫</mo>\n<mi>n</mi>\n</mrow>\n$$ g(n)\\gg n $$</annotation>\n</semantics></math> then with high probability <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0024\" display=\"inline\" location=\"graphic/rsa21199-math-0024.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msub>\n<mrow>\n<mi>R</mi>\n</mrow>\n<mrow>\n<mi>n</mi>\n</mrow>\n</msub>\n</mrow>\n$$ {R}_n $$</annotation>\n</semantics></math> is connected. These results also hold when we consider orientable and non-orientable surfaces separately. We also investigate random graphs sampled uniformly from the ‘hereditary part’ or the ‘minor-closed part’ of <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0025\" display=\"inline\" location=\"graphic/rsa21199-math-0025.png\" overflow=\"scroll\">\n<semantics>\n<mrow>\n<msup>\n<mrow>\n<mi mathvariant=\"script\">E</mi>\n</mrow>\n<mrow>\n<mi>g</mi>\n</mrow>\n</msup>\n</mrow>\n$$ {\\mathcal{E}}^g $$</annotation>\n</semantics></math>, and briefly consider corresponding results for unlabelled graphs.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"1 11-12","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures and Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/rsa.21199","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 3

Abstract

Given a ‘genus function’ g = g ( n ) $$ g=g(n) $$ , we let E g $$ {\mathcal{E}}^g $$ be the class of all graphs G $$ G $$ such that if G $$ G $$ has order n $$ n $$ (i.e., has n $$ n $$ vertices) then it is embeddable in a surface of Euler genus at most g ( n ) $$ g(n) $$ . Let the random graph R n $$ {R}_n $$ be sampled uniformly from the graphs in E g $$ {\mathcal{E}}^g $$ on vertex set [ n ] = { 1 , , n } $$ \left[n\right]=\left\{1,\dots, n\right\} $$ . Observe that if g ( n ) $$ g(n) $$ is 0 then R n $$ {R}_n $$ is a random planar graph, and if g ( n ) $$ g(n) $$ is sufficiently large then R n $$ {R}_n $$ is a binomial random graph G ( n , 1 2 ) $$ G\left(n,\frac{1}{2}\right) $$ . We investigate typical properties of R n $$ {R}_n $$ . We find that for every genus function g $$ g $$ , with high probability at most one component of R n $$ {R}_n $$ is non-planar. In contrast, we find a transition for example for connectivity: if g ( n ) $$ g(n) $$ is O ( n / log n ) $$ O\left(n/\log n\right) $$ and g $$ g $$ is non-decreasing then lim inf n ( R n is connected ) < 1 $$ \lim\ {\operatorname{inf}}_{n\to \infty}\mathbb{P}\left({R}_n\kern0.3em \mathrm{is}\ \mathrm{connected}\right)<1 $$ , and if g ( n ) n $$ g(n)\gg n $$ then with high probability R n $$ {R}_n $$ is connected. These results also hold when we consider orientable and non-orientable surfaces separately. We also investigate random graphs sampled uniformly from the ‘hereditary part’ or the ‘minor-closed part’ of E g $$ {\mathcal{E}}^g $$ , and briefly consider corresponding results for unlabelled graphs.
可嵌入顺序相关曲面的随机图
给定一个“格函数”g=g(n) $$ g=g(n) $$,我们设Eg $$ {\mathcal{E}}^g $$为所有图g $$ G $$的类,使得如果g $$ G $$有n阶$$ n $$(即,有n $$ n $$个顶点),那么它最多可嵌入到g(n) $$ g(n) $$的欧拉格曲面中。设随机图Rn $$ {R}_n $$在顶点集[n]=,n $$ \left[n\right]=\left\{1,\dots, n\right\} $$上从Eg {}$$ {\mathcal{E}}^g $$中的图中均匀采样。观察到,如果g(n) $$ g(n) $$为0,则Rn $$ {R}_n $$是一个随机平面图,如果g(n) $$ g(n) $$足够大,则Rn $$ {R}_n $$是一个二项随机图g(n,12) $$ G\left(n,\frac{1}{2}\right) $$。我们研究了Rn $$ {R}_n $$的典型性质。我们发现,对于每一个格函数g $$ g $$,在高概率下Rn $$ {R}_n $$最多有一个分量是非平面的。相反,我们发现了一个过渡,例如连通性:如果g(n) $$ g(n) $$是O(n/logn) $$ O\left(n/\log n\right) $$并且g $$ g $$是非递减的,那么lim infn→∞(Rnis connected)&lt;1 $$ \lim\ {\operatorname{inf}}_{n\to \infty}\mathbb{P}\left({R}_n\kern0.3em \mathrm{is}\ \mathrm{connected}\right)<1 $$,如果g(n) n $$ g(n)\gg n $$那么高概率Rn $$ {R}_n $$是连通的。当我们分别考虑可定向和不可定向表面时,这些结果也成立。我们还研究了从Eg $$ {\mathcal{E}}^g $$的“遗传部分”或“小闭部分”均匀抽样的随机图,并简要考虑了未标记图的相应结果。
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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