Improved Upper Bounds for Gallai–Ramsey Numbers of Paths and Cycles. Issue 1 (31st January 2013)

• Record Type: Journal Article
• Authors: Hall, Martin; Magnant, Colton; Ozeki, Kenta; Tsugaki, Masao
• Is Part Of: Journal of graph theory. Volume 75:Issue 1(2014)
• Page Start: 59
• Page End: 74
• Publication Date: 2013-01-31
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21723
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Given a graph <italic>G</italic> and a positive integer <italic>k</italic>, define the Gallai–Ramsey number to be the minimum number of vertices <italic>n</italic> such that any <italic>k</italic>‐edge coloring of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1p37mh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:jgt21723:equation:jgt21723-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> contains either a rainbow (all different colored) triangle or a monochromatic copy of <italic>G</italic>. In this work, we improve upon known upper bounds on the Gallai–Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of magnitude as functions of <italic>k</italic>.</p> </abstract>