弱饱和随机图

Zsolt Bartha, Brett Kolesnik
{"title":"弱饱和随机图","authors":"Zsolt Bartha, Brett Kolesnik","doi":"10.1002/rsa.21210","DOIUrl":null,"url":null,"abstract":"As introduced by Bollobás, a graph <mjx-container aria-label=\"upper G\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/11561f75-127e-4f60-9ceb-e40378b00f88/rsa21210-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\">G</mi></mrow>$$ G $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is weakly <mjx-container aria-label=\"upper H\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/19810fb2-a72c-4618-af37-9f48b522a70f/rsa21210-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-saturated if the complete graph <mjx-container aria-label=\"upper K Subscript n\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper K Subscript n\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.04em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/7304fecf-1a6e-4442-bc4e-9ad83100d4b9/rsa21210-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper K Subscript n\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">K</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></msub></mrow>$$ {K}_n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is obtained by iteratively completing copies of <mjx-container aria-label=\"upper H\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/4631816e-b7d6-4204-b80e-aae9d6c08f20/rsa21210-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> minus an edge. For all graphs <mjx-container aria-label=\"upper H\" ctxtmenu_counter=\"4\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/aec60805-789a-48d9-bb9e-7b2070463fc2/rsa21210-math-0005.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, we obtain an asymptotic lower bound for the critical threshold <mjx-container aria-label=\"p Subscript c\" ctxtmenu_counter=\"5\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"p Subscript c\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/2063a2ca-359a-4b1f-99c3-3aaacb0da85a/rsa21210-math-0006.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"p Subscript c\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">p</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">c</mi></mrow></msub></mrow>$$ {p}_c $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, at which point the Erdős–Rényi graph <mjx-container aria-label=\"script upper G Subscript n comma p\" ctxtmenu_counter=\"6\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-mrow><mjx-msub data-semantic-children=\"0,4\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"script upper G Subscript n comma p\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"script\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.128em;\"><mjx-mrow data-semantic-children=\"1,2,3\" data-semantic-content=\"2\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\" size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"4\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/cc83880d-e018-4e21-9c20-c2727b3024ff/rsa21210-math-0007.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,4\" data-semantic-role=\"latinletter\" data-semantic-speech=\"script upper G Subscript n comma p\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"script\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">𝒢</mi></mrow><mrow data-semantic-=\"\" data-semantic-children=\"1,2,3\" data-semantic-content=\"2\" data-semantic-parent=\"5\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"4\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">p</mi></mrow></msub></mrow></math></mjx-assistive-mml></mjx-container> is likely to be weakly <mjx-container aria-label=\"upper H\" ctxtmenu_counter=\"7\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/485c2681-0422-4e93-bbe6-41a29ca642a0/rsa21210-math-0008.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-saturated. We also prove an upper bound for <mjx-container aria-label=\"p Subscript c\" ctxtmenu_counter=\"8\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"p Subscript c\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/70b3a74f-10b2-4d96-95e1-8de139e354a6/rsa21210-math-0009.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"p Subscript c\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">p</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">c</mi></mrow></msub></mrow>$$ {p}_c $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, for all <mjx-container aria-label=\"upper H\" ctxtmenu_counter=\"9\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/876d319e-c9a0-4c80-aa1c-6a4084a04445/rsa21210-math-0010.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper H\" data-semantic-type=\"identifier\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> which are, in a sense, strictly balanced. In particular, we improve the upper bound by Balogh, Bollobás, and Morris for <mjx-container aria-label=\"upper H equals upper K Subscript r\" ctxtmenu_counter=\"10\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,4\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"upper H equals upper K Subscript r\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"5\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"2,3\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.04em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/0a578499-65cf-4474-bcb1-4576eb13ba1e/rsa21210-math-0011.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,4\" data-semantic-content=\"1\" data-semantic-role=\"equality\" data-semantic-speech=\"upper H equals upper K Subscript r\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">H</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"5\" data-semantic-role=\"equality\" data-semantic-type=\"relation\">=</mo><msub data-semantic-=\"\" data-semantic-children=\"2,3\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">K</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">r</mi></mrow></msub></mrow>$$ H={K}_r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, and we conjecture that this is sharp up to constants.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"25 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Weakly saturated random graphs\",\"authors\":\"Zsolt Bartha, Brett Kolesnik\",\"doi\":\"10.1002/rsa.21210\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"As introduced by Bollobás, a graph <mjx-container aria-label=\\\"upper G\\\" ctxtmenu_counter=\\\"0\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper G\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/11561f75-127e-4f60-9ceb-e40378b00f88/rsa21210-math-0001.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper G\\\" data-semantic-type=\\\"identifier\\\">G</mi></mrow>$$ G $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is weakly <mjx-container aria-label=\\\"upper H\\\" ctxtmenu_counter=\\\"1\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/19810fb2-a72c-4618-af37-9f48b522a70f/rsa21210-math-0002.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-saturated if the complete graph <mjx-container aria-label=\\\"upper K Subscript n\\\" ctxtmenu_counter=\\\"2\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper K Subscript n\\\" data-semantic-type=\\\"subscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: -0.15em; margin-left: -0.04em;\\\"><mjx-mrow size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/7304fecf-1a6e-4442-bc4e-9ad83100d4b9/rsa21210-math-0003.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper K Subscript n\\\" data-semantic-type=\\\"subscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">K</mi></mrow><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow></msub></mrow>$$ {K}_n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is obtained by iteratively completing copies of <mjx-container aria-label=\\\"upper H\\\" ctxtmenu_counter=\\\"3\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/4631816e-b7d6-4204-b80e-aae9d6c08f20/rsa21210-math-0004.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> minus an edge. For all graphs <mjx-container aria-label=\\\"upper H\\\" ctxtmenu_counter=\\\"4\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/aec60805-789a-48d9-bb9e-7b2070463fc2/rsa21210-math-0005.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, we obtain an asymptotic lower bound for the critical threshold <mjx-container aria-label=\\\"p Subscript c\\\" ctxtmenu_counter=\\\"5\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"p Subscript c\\\" data-semantic-type=\\\"subscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mrow size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/2063a2ca-359a-4b1f-99c3-3aaacb0da85a/rsa21210-math-0006.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"p Subscript c\\\" data-semantic-type=\\\"subscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">p</mi></mrow><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">c</mi></mrow></msub></mrow>$$ {p}_c $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, at which point the Erdős–Rényi graph <mjx-container aria-label=\\\"script upper G Subscript n comma p\\\" ctxtmenu_counter=\\\"6\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-mrow><mjx-msub data-semantic-children=\\\"0,4\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"script upper G Subscript n comma p\\\" data-semantic-type=\\\"subscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"script\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: -0.15em; margin-left: -0.128em;\\\"><mjx-mrow data-semantic-children=\\\"1,2,3\\\" data-semantic-content=\\\"2\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\" size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\" rspace=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/cc83880d-e018-4e21-9c20-c2727b3024ff/rsa21210-math-0007.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mrow><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,4\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"script upper G Subscript n comma p\\\" data-semantic-type=\\\"subscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"script\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">𝒢</mi></mrow><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"1,2,3\\\" data-semantic-content=\\\"2\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\">,</mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">p</mi></mrow></msub></mrow></math></mjx-assistive-mml></mjx-container> is likely to be weakly <mjx-container aria-label=\\\"upper H\\\" ctxtmenu_counter=\\\"7\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/485c2681-0422-4e93-bbe6-41a29ca642a0/rsa21210-math-0008.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-saturated. We also prove an upper bound for <mjx-container aria-label=\\\"p Subscript c\\\" ctxtmenu_counter=\\\"8\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"p Subscript c\\\" data-semantic-type=\\\"subscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mrow size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/70b3a74f-10b2-4d96-95e1-8de139e354a6/rsa21210-math-0009.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"p Subscript c\\\" data-semantic-type=\\\"subscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">p</mi></mrow><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">c</mi></mrow></msub></mrow>$$ {p}_c $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, for all <mjx-container aria-label=\\\"upper H\\\" ctxtmenu_counter=\\\"9\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/876d319e-c9a0-4c80-aa1c-6a4084a04445/rsa21210-math-0010.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper H\\\" data-semantic-type=\\\"identifier\\\">H</mi></mrow>$$ H $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> which are, in a sense, strictly balanced. In particular, we improve the upper bound by Balogh, Bollobás, and Morris for <mjx-container aria-label=\\\"upper H equals upper K Subscript r\\\" ctxtmenu_counter=\\\"10\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,4\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-role=\\\"equality\\\" data-semantic-speech=\\\"upper H equals upper K Subscript r\\\" data-semantic-type=\\\"relseq\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"relseq,=\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"equality\\\" data-semantic-type=\\\"relation\\\" rspace=\\\"5\\\" space=\\\"5\\\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\\\"2,3\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: -0.15em; margin-left: -0.04em;\\\"><mjx-mrow size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/0a578499-65cf-4474-bcb1-4576eb13ba1e/rsa21210-math-0011.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,4\\\" data-semantic-content=\\\"1\\\" data-semantic-role=\\\"equality\\\" data-semantic-speech=\\\"upper H equals upper K Subscript r\\\" data-semantic-type=\\\"relseq\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">H</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"relseq,=\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"equality\\\" data-semantic-type=\\\"relation\\\">=</mo><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"2,3\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">K</mi></mrow><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"4\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">r</mi></mrow></msub></mrow>$$ H={K}_r $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, and we conjecture that this is sharp up to constants.\",\"PeriodicalId\":20948,\"journal\":{\"name\":\"Random Structures and Algorithms\",\"volume\":\"25 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-02-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Random Structures and Algorithms\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1002/rsa.21210\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures and Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/rsa.21210","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

正如波尔洛巴斯(Bollobás)所介绍的,如果完整图 Kn$$ {K}_n$ 是通过迭代补全 H$$ H$ 的副本减去一条边得到的,则图 G$$ G$$ 是弱 H$$ H$ 饱和的。对于所有图 H$$ H $$,我们得到了临界阈值 pc$$ {p}_c $$的渐近下限,此时厄尔多斯-雷尼图 𝒢n,p 很可能弱 H$$ H $$饱和。我们还证明了 pc$$ {p}_c $$ 的上界,适用于所有 H$$ H $$,从某种意义上说,它们都是严格平衡的。特别是,我们改进了巴洛格、波尔洛巴斯和莫里斯对 H=Kr$$ H={K}_r$ 的上界,并且我们猜想这个上界在常数以内都是尖锐的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Weakly saturated random graphs
As introduced by Bollobás, a graph is weakly -saturated if the complete graph is obtained by iteratively completing copies of minus an edge. For all graphs , we obtain an asymptotic lower bound for the critical threshold , at which point the Erdős–Rényi graph is likely to be weakly -saturated. We also prove an upper bound for , for all which are, in a sense, strictly balanced. In particular, we improve the upper bound by Balogh, Bollobás, and Morris for , and we conjecture that this is sharp up to constants.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信