A Note on Circular Chromatic Number of Graphs with Large Girth and Similar Problems. Issue 4 (23rd December 2014)

• Record Type: Journal Article
• Authors: Nešetřil, Jaroslav; Ossona de Mendez, Patrice
• Is Part Of: Journal of graph theory. Volume 80:Issue 4(2015)
• Page Start: 268
• Page End: 276
• Publication Date: 2014-12-23
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21849
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>In this short note, we extend the result of Galluccio, Goddyn, and Hell, which states that graphs of large girth excluding a minor are nearly bipartite. We also prove a similar result for the oriented chromatic number, from which follows in particular that graphs of large girth excluding a minor have oriented chromatic number at most 5, and for the <italic>p</italic>th chromatic number <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkfp3xj44" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21849:jgt21849-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>χ</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math></alternatives></inline-formula>, from which follows in particular that graphs <italic>G</italic> of large girth excluding a minor have <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkfp3xjb7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21849:jgt21849-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>χ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>.</p> </abstract>