Claw‐Free Graphs, Skeletal Graphs, and a Stronger Conjecture on ω, Δ, and χ. Issue 3 (16th April 2014)

• Record Type: Journal Article
• Authors: King, Andrew D.; Reed, Bruce A.
• Is Part Of: Journal of graph theory. Volume 78:Issue 3(2015)
• Page Start: 157
• Page End: 194
• Publication Date: 2014-04-16
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21797
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The second author's (B.A.R.) ω, Δ, χ conjecture proposes that every graph satisfies <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh30vzg59j" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21797:jgt21797-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>χ</mml:mi><mml:mo>≤</mml:mo><mml:mo>⌈</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</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:mi>ω</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>⌉</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. In this article, we prove that the conjecture holds for all claw‐free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way, we discuss a stronger local conjecture, and prove that it holds for claw‐free graphs with a three‐colorable complement. To prove our results, we introduce a very useful χ‐preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so‐called <italic>skeletal</italic> graphs.</p> </abstract>