Note on the Pair-crossing Number and the Odd-crossing Number.
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| Title: | Note on the Pair-crossing Number and the Odd-crossing Number. |
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| Authors: | Géza Tóth1 |
| Source: | Discrete & Computational Geometry. Jun2008, Vol. 39 Issue 4, p791-799. 9p. 2 Diagrams, 2 Graphs, 1 Map. |
| Subjects: | Mathematics, Writing of numerals, Roman numerals, Numerals |
| Abstract: | Abstract The crossing number [ABSTRACT FROM AUTHOR] of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number  is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number  is the minimum number of pairs of edges that cross an odd number of times. Clearly,  . We construct graphs with  . This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte. Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k 2/log 2 k). |
| Copyright of Discrete & Computational Geometry is the property of Springer Nature 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 |
| FullText | Links: – Type: pdflink Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 33051489 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Note on the Pair-crossing Number and the Odd-crossing Number. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Géza+Tóth%22">Géza Tóth</searchLink><relatesTo>1</relatesTo> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Jun2008, Vol. 39 Issue 4, p791-799. 9p. 2 Diagrams, 2 Graphs, 1 Map. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Mathematics%22">Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Writing+of+numerals%22">Writing of numerals</searchLink><br /><searchLink fieldCode="DE" term="%22Roman+numerals%22">Roman numerals</searchLink><br /><searchLink fieldCode="DE" term="%22Numerals%22">Numerals</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: <div class="Abstract"><a name="Abs1"></a><span class="AbstractHeading">Abstract  </span> The crossing number <a name="IEq1"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq1.gif" alt="${\mbox{\sc cr}}(G)$" align="middle" border="0"> of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number <a name="IEq2"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq2.gif" alt="${\mbox{\sc pair-cr}}(G)$" align="middle" border="0"> is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number <a name="IEq3"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq3.gif" alt="${\mbox{\sc odd-cr}}(G)$" align="middle" border="0"> is the minimum number of pairs of edges that cross an odd number of times. Clearly, <a name="IEq4"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq4.gif" alt="${\mbox{\sc odd-cr}}(G)\le {\mbox{\sc pair-cr}}(G)\le {\mbox{\sc cr}}(G)$" align="middle" border="0"> . We construct graphs with <a name="IEq5"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq5.gif" alt="$0.855\cdot {\mbox{\sc pair-cr}}(G)\ge {\mbox{\sc odd-cr}}(G)$" align="middle" border="0"> . This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte. <div class="AbstractPara"> <div class=""> Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k 2/log 2 k). </div> </div> </div> [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature 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: Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 9 StartPage: 791 Subjects: – SubjectFull: Mathematics Type: general – SubjectFull: Writing of numerals Type: general – SubjectFull: Roman numerals Type: general – SubjectFull: Numerals Type: general Titles: – TitleFull: Note on the Pair-crossing Number and the Odd-crossing Number. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Géza Tóth IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2008 Type: published Y: 2008 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 39 – Type: issue Value: 4 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
| ResultId | 1 |
