On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors. Issue 2 (29th May 2012)

• Record Type: Journal Article
• Authors: Bryant, Darryn; Danziger, Peter; Dean, Matthew
• Is Part Of: Journal of combinatorial designs. Volume 21:Issue 2(2013:Feb.)
• Page Start: 60
• Page End: 80
• Publication Date: 2012-05-29
• Subjects: Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21312
• Languages: English
• ISSNs: 1063-8539

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Given two 2‐regular graphs <italic>F</italic><sub>1</sub> and <italic>F</italic><sub>2</sub>, both of order <italic>n</italic>, the Hamilton‐Waterloo Problem for <italic>F</italic><sub>1</sub> and <italic>F</italic><sub>2</sub> asks for a factorization of the complete graph <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1n44wbgf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21312:equation:jcd21312-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> into α<sub>1</sub> copies of <italic>F</italic><sub>1</sub>, α<sub>2</sub> copies of <italic>F</italic><sub>2</sub>, and a 1‐factor if <italic>n</italic> is even, for all nonnegative integers α<sub>1</sub> and α<sub>2</sub> satisfying <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1n44wbdb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21312:equation:jcd21312-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:mfrac><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>⌋</mml:mo></mml:mrow></mml:mrow></mml:math>. We settle the Hamilton‐Waterloo Problem for all bipartite 2‐regular graphs <italic>F</italic><sub>1</sub> and <italic>F</italic><sub>2</sub> where <italic>F</italic><sub>1</sub> can be obtained from <italic>F</italic><sub>2</sub> by replacing each cycle with a bipartite 2‐regular graph of the same order.</p> </abstract>