A Note on Hitting Maximum and Maximal Cliques With a Stable Set. Issue 3 (17th July 2012)

• Record Type: Journal Article
• Authors: Christofides, Demetres; Edwards, Katherine; King, Andrew D.
• Is Part Of: Journal of graph theory. Volume 73:Issue 3(2013)
• Page Start: 354
• Page End: 360
• Publication Date: 2012-07-17
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21684
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>It was recently proved that any graph satisfying <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1zv4q81x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21684:jgt21684-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ω</mml:mi><mml:mo>></mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mi>Δ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> contains a stable set hitting every maximum clique. In this note, we prove that the same is true for graphs satisfying <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1zv4q7z9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21684:jgt21684-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ω</mml:mi><mml:mo>≥</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mi>Δ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> unless the graph is the strong product of an odd hole and <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1zv4q7w6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21684:jgt21684-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mi>ω</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>.We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.</p> </abstract>