Counterexamples to the List Square Coloring Conjecture. Issue 4 (14th April 2014)

• Record Type: Journal Article
• Authors: Kim, Seog‐Jin; Park, Boram
• Is Part Of: Journal of graph theory. Volume 78:Issue 4(2015)
• Page Start: 239
• Page End: 247
• Publication Date: 2014-04-14
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21802
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The square <italic>G</italic><sup>2</sup> of a graph <italic>G</italic> is the graph defined on <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjf863zk8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21802:jgt21802-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> such that two vertices <italic>u</italic> and <italic>v</italic> are adjacent in <italic>G</italic><sup>2</sup> if the distance between <italic>u</italic> and <italic>v</italic> in <italic>G</italic> is at most 2. Let <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjf863zpt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21802:jgt21802-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>χ</mml:mi><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjf863zf6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21802:jgt21802-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>χ</mml:mi><mml:mi>ℓ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula> be the chromatic number and the list chromatic number of a graph <italic>H</italic>, respectively. A graph <italic>H</italic> is called <italic>chromatic‐choosable</italic> if <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjf863zgq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21802:jgt21802-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>χ</mml:mi><mml:mi>ℓ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>χ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>. It is an interesting problem to find graphs that are chromatic‐choosable. Kostochka and Woodall (Choosability conjectures and multicircuits, Discrete Math., 240 (2001), 123–143) conjectured that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjf86408j" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21802:jgt21802-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>χ</mml:mi><mml:mi>ℓ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>G</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>χ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>G</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula> for every graph <italic>G</italic>, which is called List Square Coloring Conjecture. In this article, we give infinitely many counter examples to the conjecture. Moreover, we show that the value <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjf864092" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21802:jgt21802-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>χ</mml:mi><mml:mi>ℓ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>G</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>χ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>G</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula> can be arbitrarily large.</p> </abstract>