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Proof of a conjecture on fractional Ramsey numbers

✍ Scribed by Jason Brown; Richard Hoshino

Publisher: John Wiley and Sons
Year: 2009
Tongue: English
Weight: 144 KB
Volume: 63
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.20416

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✦ Synopsis

Abstract

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r~f~ (a~1~, a~2~, …, a~k~) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k=2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010

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