The minimum size of graphs satisfying cut conditions.
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| Title: | The minimum size of graphs satisfying cut conditions. |
|---|---|
| Authors: | Jobson, Adam S.1, Kézdy, André E.1, Lehel, Jenő1,2 j0lehe01@louisville.edu |
| Source: | Discrete Applied Mathematics. Mar2018, Vol. 237, p89-96. 8p. |
| Subjects: | Graph theory, Paths & cycles in graph theory, Topological degree, Mathematical bounds, Bisectors (Geometry) |
| Abstract: | A graph G of order n satisfies the cut condition (CC) if there are at least | A | edges between any set A ⊂ V ( G ) , | A | ≤ n ∕ 2 , and its complement A ¯ = V ( G ) ∖ A . For even n , G satisfies the even cut condition (ECC), if [ A , A ¯ ] contains at least n ∕ 2 edges, for every A ⊂ V ( G ) , | A | = n ∕ 2 . We investigate here the minimum number of edges in a graph G satisfying CC or ECC. A simple counting argument shows that for both cut conditions | E ( G ) | ≥ n − 1 , and the star K 1 , n − 1 is extremal. Faudree et al. (1999) conjectured that the extremal graphs with maximum degree Δ ( G ) < n − 1 satisfying ECC have 3 n ∕ 2 − O ( 1 ) edges. Here we prove the tight bound | E ( G ) | ≥ 3 n ∕ 2 − 3 , for every graph G with Δ ( G ) < n − 1 and satisfying CC. If G is 2-connected and satisfies ECC, we prove that | E ( G ) | ≥ 3 n ∕ 2 − 2 holds and tight, for every even n . We obtain the weaker bound | E ( G ) | ≥ 5 n ∕ 4 − 2 , for every graph of order n ≡ 0 ( mod 4 ) with Δ ( G ) < n − 1 and satisfying ECC; meanwhile we conjecture that | E ( G ) | ≥ 3 n ∕ 2 − 4 holds, for every even n . [ABSTRACT FROM AUTHOR] |
| Copyright of Discrete Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 128002258 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: The minimum size of graphs satisfying cut conditions. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Jobson%2C+Adam+S%2E%22">Jobson, Adam S.</searchLink><relatesTo>1</relatesTo><br /><searchLink fieldCode="AR" term="%22Kézdy%2C+André+E%2E%22">Kézdy, André E.</searchLink><relatesTo>1</relatesTo><br /><searchLink fieldCode="AR" term="%22Lehel%2C+Jenő%22">Lehel, Jenő</searchLink><relatesTo>1,2</relatesTo><i> j0lehe01@louisville.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+Applied+Mathematics%22">Discrete Applied Mathematics</searchLink>. Mar2018, Vol. 237, p89-96. 8p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Graph+theory%22">Graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Paths+%26+cycles+in+graph+theory%22">Paths & cycles in graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Topological+degree%22">Topological degree</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+bounds%22">Mathematical bounds</searchLink><br /><searchLink fieldCode="DE" term="%22Bisectors+%28Geometry%29%22">Bisectors (Geometry)</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: A graph G of order n satisfies the cut condition (CC) if there are at least | A | edges between any set A ⊂ V ( G ) , | A | ≤ n ∕ 2 , and its complement A ¯ = V ( G ) ∖ A . For even n , G satisfies the even cut condition (ECC), if [ A , A ¯ ] contains at least n ∕ 2 edges, for every A ⊂ V ( G ) , | A | = n ∕ 2 . We investigate here the minimum number of edges in a graph G satisfying CC or ECC. A simple counting argument shows that for both cut conditions | E ( G ) | ≥ n − 1 , and the star K 1 , n − 1 is extremal. Faudree et al. (1999) conjectured that the extremal graphs with maximum degree Δ ( G ) < n − 1 satisfying ECC have 3 n ∕ 2 − O ( 1 ) edges. Here we prove the tight bound | E ( G ) | ≥ 3 n ∕ 2 − 3 , for every graph G with Δ ( G ) < n − 1 and satisfying CC. If G is 2-connected and satisfies ECC, we prove that | E ( G ) | ≥ 3 n ∕ 2 − 2 holds and tight, for every even n . We obtain the weaker bound | E ( G ) | ≥ 5 n ∕ 4 − 2 , for every graph of order n ≡ 0 ( mod 4 ) with Δ ( G ) < n − 1 and satisfying ECC; meanwhile we conjecture that | E ( G ) | ≥ 3 n ∕ 2 − 4 holds, for every even n . [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Discrete Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.dam.2017.11.014 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 8 StartPage: 89 Subjects: – SubjectFull: Graph theory Type: general – SubjectFull: Paths & cycles in graph theory Type: general – SubjectFull: Topological degree Type: general – SubjectFull: Mathematical bounds Type: general – SubjectFull: Bisectors (Geometry) Type: general Titles: – TitleFull: The minimum size of graphs satisfying cut conditions. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Jobson, Adam S. – PersonEntity: Name: NameFull: Kézdy, André E. – PersonEntity: Name: NameFull: Lehel, Jenő IsPartOfRelationships: – BibEntity: Dates: – D: 11 M: 03 Text: Mar2018 Type: published Y: 2018 Identifiers: – Type: issn-print Value: 0166218X Numbering: – Type: volume Value: 237 Titles: – TitleFull: Discrete Applied Mathematics Type: main |
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