多和弦循环

Nemanja Draganić, Abhishek Methuku, David Munhá Correia, Benny Sudakov
{"title":"多和弦循环","authors":"Nemanja Draganić, Abhishek Methuku, David Munhá Correia, Benny Sudakov","doi":"10.1002/rsa.21207","DOIUrl":null,"url":null,"abstract":"How many edges in an <mjx-container aria-label=\"Menu available. Press control and space , or space\" 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\" location=\"graphic/rsa21207-math-0001.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" 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=\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0001\" display=\"inline\" location=\"graphic/rsa21207-math-0001.png\" overflow=\"scroll\" 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=\"n\" data-semantic-type=\"identifier\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any <mjx-container aria-label=\"Menu available. Press control and space , or space\" 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\" location=\"graphic/rsa21207-math-0002.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" 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=\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0002\" display=\"inline\" location=\"graphic/rsa21207-math-0002.png\" overflow=\"scroll\" 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=\"n\" data-semantic-type=\"identifier\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-vertex graph with <mjx-container aria-label=\"Menu available. Press control and space , or space\" 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\" location=\"graphic/rsa21207-math-0003.png\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,6\" data-semantic-content=\"7\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"2 n Superscript 3 divided by 2\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"8\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\"1,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"latinletter\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow data-semantic-children=\"2,4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"division\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"5\" data-semantic-role=\"division\" data-semantic-type=\"operator\" rspace=\"1\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0003\" display=\"inline\" location=\"graphic/rsa21207-math-0003.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,6\" data-semantic-content=\"7\" data-semantic-role=\"implicit\" data-semantic-speech=\"2 n Superscript 3 divided by 2\" data-semantic-type=\"infixop\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"8\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"8\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\">⁢</mo><msup data-semantic-=\"\" data-semantic-children=\"1,5\" data-semantic-parent=\"8\" data-semantic-role=\"latinletter\" data-semantic-type=\"superscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow><mrow data-semantic-=\"\" data-semantic-children=\"2,4\" data-semantic-content=\"3\" data-semantic-parent=\"6\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn><mo data-semantic-=\"\" data-semantic-operator=\"infixop,/\" data-semantic-parent=\"5\" data-semantic-role=\"division\" data-semantic-type=\"operator\" stretchy=\"false\">/</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></mrow></msup></mrow>$$ 2{n}^{3/2} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> edges contains such a cycle. We significantly improve this old bound by showing that <mjx-container aria-label=\"Menu available. Press control and space , or space\" 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\" location=\"graphic/rsa21207-math-0004.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,13\" data-semantic-content=\"14,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"normal upper Omega left parenthesis n log Superscript 8 Baseline n right parenthesis\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"12\" data-semantic-content=\"1,8\" data-semantic- data-semantic-parent=\"15\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"2,10\" data-semantic-content=\"11\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mspace style=\"width: 0.2em;\"></mjx-mspace><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"6,7\" data-semantic-content=\"9,4\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mjx-msup data-semantic-children=\"4,5\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"6\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.421em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0004\" display=\"inline\" location=\"graphic/rsa21207-math-0004.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,13\" data-semantic-content=\"14,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"normal upper Omega left parenthesis n log Superscript 8 Baseline n right parenthesis\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\" mathvariant=\"normal\">Ω</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\">⁡</mo><mrow data-semantic-=\"\" data-semantic-children=\"12\" data-semantic-content=\"1,8\" data-semantic-parent=\"15\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"2,10\" data-semantic-content=\"11\" data-semantic-parent=\"13\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mspace width=\"0.2em\"></mspace><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\">⁢</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"6,7\" data-semantic-content=\"9,4\" data-semantic-parent=\"12\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><msup data-semantic-=\"\" data-semantic-children=\"4,5\" data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"superscript\"><mrow><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-operator=\"appl\" data-semantic-parent=\"6\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\">log</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">8</mn></mrow></msup><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\">⁡</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow>$$ \\Omega \\left(n\\kern0.2em {\\log}^8n\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> edges are enough to guarantee the existence of such a cycle. Our proof exploits a delicate interplay between certain properties of random walks in almost regular expanders. We argue that while the probability that a random walk of certain length in an almost regular expander is self-avoiding is very small, one can still guarantee that it spans many edges (and that it can be closed into a cycle) with large enough probability to ensure that these two events happen simultaneously.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Cycles with many chords\",\"authors\":\"Nemanja Draganić, Abhishek Methuku, David Munhá Correia, Benny Sudakov\",\"doi\":\"10.1002/rsa.21207\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"How many edges in an <mjx-container aria-label=\\\"Menu available. Press control and space , or space\\\" 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\\\" location=\\\"graphic/rsa21207-math-0001.png\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"n\\\" 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=\\\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0001\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21207-math-0001.png\\\" overflow=\\\"scroll\\\" 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=\\\"n\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any <mjx-container aria-label=\\\"Menu available. Press control and space , or space\\\" 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\\\" location=\\\"graphic/rsa21207-math-0002.png\\\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"n\\\" 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=\\\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0002\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21207-math-0002.png\\\" overflow=\\\"scroll\\\" 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=\\\"n\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-vertex graph with <mjx-container aria-label=\\\"Menu available. Press control and space , or space\\\" 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\\\" location=\\\"graphic/rsa21207-math-0003.png\\\"><mjx-semantics><mjx-mrow data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"0,6\\\" data-semantic-content=\\\"7\\\" data-semantic- data-semantic-role=\\\"implicit\\\" data-semantic-speech=\\\"2 n Superscript 3 divided by 2\\\" data-semantic-type=\\\"infixop\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"infixop,⁢\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\\\"1,5\\\" data-semantic- data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"superscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: 0.363em;\\\"><mjx-mrow data-semantic-children=\\\"2,4\\\" data-semantic-content=\\\"3\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"infixop\\\" size=\\\"s\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\" space=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0003\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21207-math-0003.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"0,6\\\" data-semantic-content=\\\"7\\\" data-semantic-role=\\\"implicit\\\" data-semantic-speech=\\\"2 n Superscript 3 divided by 2\\\" data-semantic-type=\\\"infixop\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">2</mn><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"infixop,⁢\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\">⁢</mo><msup data-semantic-=\\\"\\\" data-semantic-children=\\\"1,5\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"superscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"2,4\\\" data-semantic-content=\\\"3\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"infixop\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">3</mn><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\" stretchy=\\\"false\\\">/</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">2</mn></mrow></msup></mrow>$$ 2{n}^{3/2} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> edges contains such a cycle. We significantly improve this old bound by showing that <mjx-container aria-label=\\\"Menu available. Press control and space , or space\\\" 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\\\" location=\\\"graphic/rsa21207-math-0004.png\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,13\\\" data-semantic-content=\\\"14,0\\\" data-semantic- data-semantic-role=\\\"simple function\\\" data-semantic-speech=\\\"normal upper Omega left parenthesis n log Superscript 8 Baseline n right parenthesis\\\" data-semantic-type=\\\"appl\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"15\\\" data-semantic-role=\\\"simple function\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"15\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\\\"12\\\" data-semantic-content=\\\"1,8\\\" data-semantic- data-semantic-parent=\\\"15\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"2,10\\\" data-semantic-content=\\\"11\\\" data-semantic- data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mspace style=\\\"width: 0.2em;\\\"></mjx-mspace><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"infixop,⁢\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"6,7\\\" data-semantic-content=\\\"9,4\\\" data-semantic- data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"appl\\\"><mjx-msup data-semantic-children=\\\"4,5\\\" data-semantic- data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"superscript\\\"><mjx-mrow><mjx-mi data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"function\\\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: 0.421em;\\\"><mjx-mrow size=\\\"s\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0004\\\" display=\\\"inline\\\" location=\\\"graphic/rsa21207-math-0004.png\\\" overflow=\\\"scroll\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,13\\\" data-semantic-content=\\\"14,0\\\" data-semantic-role=\\\"simple function\\\" data-semantic-speech=\\\"normal upper Omega left parenthesis n log Superscript 8 Baseline n right parenthesis\\\" data-semantic-type=\\\"appl\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"15\\\" data-semantic-role=\\\"simple function\\\" data-semantic-type=\\\"identifier\\\" mathvariant=\\\"normal\\\">Ω</mi><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"15\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\">⁡</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"12\\\" data-semantic-content=\\\"1,8\\\" data-semantic-parent=\\\"15\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">(</mo><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"2,10\\\" data-semantic-content=\\\"11\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi><mspace width=\\\"0.2em\\\"></mspace><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"infixop,⁢\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\">⁢</mo><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"6,7\\\" data-semantic-content=\\\"9,4\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"appl\\\"><msup data-semantic-=\\\"\\\" data-semantic-children=\\\"4,5\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"superscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-font=\\\"normal\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"prefix function\\\" data-semantic-type=\\\"function\\\">log</mi></mrow><mrow><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">8</mn></mrow></msup><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\">⁡</mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">)</mo></mrow></mrow>$$ \\\\Omega \\\\left(n\\\\kern0.2em {\\\\log}^8n\\\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> edges are enough to guarantee the existence of such a cycle. Our proof exploits a delicate interplay between certain properties of random walks in almost regular expanders. We argue that while the probability that a random walk of certain length in an almost regular expander is self-avoiding is very small, one can still guarantee that it spans many edges (and that it can be closed into a cycle) with large enough probability to ensure that these two events happen simultaneously.\",\"PeriodicalId\":20948,\"journal\":{\"name\":\"Random Structures and Algorithms\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-01-10\",\"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.21207\",\"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.21207","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

在一个 n$$ n$ 个顶点的图中,有多少条边会迫使存在一个具有与顶点数量相同的弦的循环?将近 30 年前,陈(Chen)、厄多斯(Erdős)和斯塔顿(Staton)考虑过这个问题,并证明任何具有 2n3/2$$ 2{n}^{3/2} $$ 条边的 n$$ n$ 个顶点图都包含这样一个循环。我们通过证明Ω(nlog8n)$$ \Omega \left(n\kern0.2em {\log}^8n\right) $$边就足以保证这样一个循环的存在,从而大大改进了这一旧界值。我们的证明利用了几乎正则扩展器中随机游走的某些特性之间微妙的相互作用。我们认为,虽然几乎正则扩展器中一定长度的随机漫步自避开的概率非常小,但我们仍然可以保证它以足够大的概率跨越许多边(并且可以闭合成一个循环),以确保这两个事件同时发生。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Cycles with many chords
How many edges in an -vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any -vertex graph with edges contains such a cycle. We significantly improve this old bound by showing that edges are enough to guarantee the existence of such a cycle. Our proof exploits a delicate interplay between certain properties of random walks in almost regular expanders. We argue that while the probability that a random walk of certain length in an almost regular expander is self-avoiding is very small, one can still guarantee that it spans many edges (and that it can be closed into a cycle) with large enough probability to ensure that these two events happen simultaneously.
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