A Sufficient Condition for Edge Chromatic Critical Graphs to Be Hamiltonian—An Approach to Vizing's 2‐Factor Conjecture. Issue 4 (11th July 2012)

• Record Type: Journal Article
• Authors: Luo, Rong; Zhao, Yue
• Is Part Of: Journal of graph theory. Volume 73:Issue 4(2013)
• Page Start: 469
• Page End: 482
• Publication Date: 2012-07-11
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21689
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>In this article, we consider Vizing's 2‐Factor Conjecture which claims that any Δ‐critical graph has a 2‐factor, and show that if <italic>G</italic> is a Δ‐critical graph with <italic>n</italic> vertices satisfying <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg21g219bd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21689:jgt21689-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Δ</mml:mi><mml:mo>≥</mml:mo><mml:mfrac><mml:mrow><mml:mn>6</mml:mn><mml:mi>n</mml:mi></mml:mrow><mml:mn>7</mml:mn></mml:mfrac></mml:mrow></mml:math>, then <italic>G</italic> is Hamiltonian and thus <italic>G</italic> has a 2‐factor. Meanwhile in this article, we also consider long cycles of overfull critical graphs and obtain that if <italic>G</italic> is an overfull Δ‐critical graph with <italic>n</italic> vertices, then the circumference of <italic>G</italic> is at least min<inline-graphic mimetype="image" xlink:href="ark:/27927/pgg21g21989" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21689:jgt21689-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mn>2</mml:mn><mml:mi>Δ</mml:mi><mml:mo>, </mml:mo><mml:mi>n</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:math>.© 2012 Wiley Periodicals, Inc. J. Graph Theory 00: 1‐14, 2012</p> </abstract>