A Tighter Erdős‐Pósa Function for Long Cycles. Issue 2 (5th December 2013)

• Record Type: Journal Article
• Authors: Fiorini, Samuel; Herinckx, Audrey
• Is Part Of: Journal of graph theory. Volume 77:Issue 2(2014)
• Page Start: 111
• Page End: 116
• Publication Date: 2013-12-05
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21776
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We prove that there exists a bivariate function <italic>f</italic> with <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghxt6thgf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21776:jgt21776-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>, </mml:mo><mml:mi>ℓ</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>·</mml:mo><mml:mi>k</mml:mi><mml:mo form="prefix">log</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> such that for every natural <italic>k</italic> and ℓ, every graph <italic>G</italic> has at least <italic>k</italic> vertex‐disjoint cycles of length at least ℓ or a set of at most <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghxt6thfx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21776:jgt21776-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>, </mml:mo><mml:mi>ℓ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> vertices that meets all cycles of length at least ℓ. This improves a result by Birmelé et al. (Combinatorica, 27 (2007), 135–145), who proved the same result with <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghxt6thjg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21776:jgt21776-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>, </mml:mo><mml:mi>ℓ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>Θ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>·</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>.</p> </abstract>