Large Monochromatic Triple Stars in Edge Colourings. Issue 4 (19th January 2015)

• Record Type: Journal Article
• Authors: Letzter, Shoham
• Is Part Of: Journal of graph theory. Volume 80:Issue 4(2015)
• Page Start: 323
• Page End: 328
• Publication Date: 2015-01-19
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21854
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Following problems posed by Gyárfás 2011, we show that for every <italic>r</italic>‐edge‐colouring of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkfp3x738" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21854:jgt21854-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula> there is a monochromatic triple star of order at least <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkfp3x717" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21854:jgt21854-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, improving Ruszinkó's result 2012. An edge colouring of a graph is called a local <italic>r</italic>‐colouring if every vertex spans edges of at most <italic>r</italic> distinct colours. We prove the existence of a monochromatic triple star with at least <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkfp3x72r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21854:jgt21854-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>r</mml:mi><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> vertices in every local <italic>r</italic>‐colouring of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkfp3x76t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21854:jgt21854-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula>.</p> </abstract>