{"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><</mo>\n<mn>1</mn>\n</mrow>\n$$ \\lim\\ {\\operatorname{inf}}_{n\\to \\infty}\\mathbb{P}\\left({R}_n\\kern0.3em \\mathrm{is}\\ \\mathrm{connected}\\right)<1 $$</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’ , we let be the class of all graphs such that if has order (i.e., has vertices) then it is embeddable in a surface of Euler genus at most . Let the random graph be sampled uniformly from the graphs in on vertex set . Observe that if is 0 then is a random planar graph, and if is sufficiently large then is a binomial random graph . We investigate typical properties of . We find that for every genus function , with high probability at most one component of is non-planar. In contrast, we find a transition for example for connectivity: if is and is non-decreasing then , and if then with high probability 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 , and briefly consider corresponding results for unlabelled graphs.