Pub Date : 2021-08-01DOI: 10.1017/9781108912181.015
V. Chvátal
{"title":"A Few Tricks of the Trade","authors":"V. Chvátal","doi":"10.1017/9781108912181.015","DOIUrl":"https://doi.org/10.1017/9781108912181.015","url":null,"abstract":"","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124203182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-01DOI: 10.1017/9781108912181.017
V. Chvátal
{"title":"More on Erdős","authors":"V. Chvátal","doi":"10.1017/9781108912181.017","DOIUrl":"https://doi.org/10.1017/9781108912181.017","url":null,"abstract":"","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"145 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120886654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-01DOI: 10.1017/9781108912181.005
V. Chvátal
{"title":"Discrete Geometry and Spinoffs","authors":"V. Chvátal","doi":"10.1017/9781108912181.005","DOIUrl":"https://doi.org/10.1017/9781108912181.005","url":null,"abstract":"","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133372862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-01DOI: 10.1017/9781108912181.013
V. Chvátal
{"title":"Thresholds of Graph Properties","authors":"V. Chvátal","doi":"10.1017/9781108912181.013","DOIUrl":"https://doi.org/10.1017/9781108912181.013","url":null,"abstract":"","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128001337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-31DOI: 10.1017/9781108912181.019
{"title":"Index","authors":"","doi":"10.1017/9781108912181.019","DOIUrl":"https://doi.org/10.1017/9781108912181.019","url":null,"abstract":"","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"74 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128398364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-31DOI: 10.1017/9781108912181.014
Frank de Zeeuw
Let us prove that it has no 10-cycle, so the circumference is 9. We think of the Petersen graph as an outside 5-cycle and an inside 5-cycle, connected by 5 links. A 10-cycle would have to contain an even number of such links, and not 0 since then we would not get a connected subgraph. Up to isomorphism, this leaves the two cases below: two links or four links (depicted with thick black edges); note that in the case of two links, they must hit adjacent vertices on one of the two 5-cycles, so after possibly swapping the 5-cycles, this is the only case with two links. In each case, we mark edges that cannot be in the cycle with red, and edges that must be in the cycle with green. Whenever a vertex has a red edge, its other two edges must be green or black. And if two edges of a vertex are green or black, then the third edge must be red. In this way we get a contradiction in both cases, either because of a vertex of degree 3 or because of a 5-cycle.
{"title":"Hamilton Cycles","authors":"Frank de Zeeuw","doi":"10.1017/9781108912181.014","DOIUrl":"https://doi.org/10.1017/9781108912181.014","url":null,"abstract":"Let us prove that it has no 10-cycle, so the circumference is 9. We think of the Petersen graph as an outside 5-cycle and an inside 5-cycle, connected by 5 links. A 10-cycle would have to contain an even number of such links, and not 0 since then we would not get a connected subgraph. Up to isomorphism, this leaves the two cases below: two links or four links (depicted with thick black edges); note that in the case of two links, they must hit adjacent vertices on one of the two 5-cycles, so after possibly swapping the 5-cycles, this is the only case with two links. In each case, we mark edges that cannot be in the cycle with red, and edges that must be in the cycle with green. Whenever a vertex has a red edge, its other two edges must be green or black. And if two edges of a vertex are green or black, then the third edge must be red. In this way we get a contradiction in both cases, either because of a vertex of degree 3 or because of a 5-cycle.","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121188488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-31DOI: 10.1017/cbo9780511803888.011
Tom Ridge
The infinite form of Ramsey’s Theorem is proved following Boolos and Jeffrey, Chapter 26.
根据Boolos和Jeffrey在第26章的论述，证明了Ramsey定理的无限形式。
{"title":"Ramsey’s Theorem","authors":"Tom Ridge","doi":"10.1017/cbo9780511803888.011","DOIUrl":"https://doi.org/10.1017/cbo9780511803888.011","url":null,"abstract":"The infinite form of Ramsey’s Theorem is proved following Boolos and Jeffrey, Chapter 26.","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126229638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-31DOI: 10.1017/9781108912181.011
E. Walker
. . . in other words, if every pair of people at a party shares exactly one mutual friend at the party then some guest is a friend of everybody present. More is is true: friendships must be arranged in triangles which intersect only in the universally popular guest. So the graphs satisfying the hypothesis of the theorem are those shown on the immediate right, usually known as ‘windmill’ graphs.
{"title":"The Friendship Theorem","authors":"E. Walker","doi":"10.1017/9781108912181.011","DOIUrl":"https://doi.org/10.1017/9781108912181.011","url":null,"abstract":". . . in other words, if every pair of people at a party shares exactly one mutual friend at the party then some guest is a friend of everybody present. More is is true: friendships must be arranged in triangles which intersect only in the universally popular guest. So the graphs satisfying the hypothesis of the theorem are those shown on the immediate right, usually known as ‘windmill’ graphs.","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121976548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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