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BROOKS’ THEOREM FOR MEASURABLE COLORINGS

Published online by Cambridge University Press:  24 June 2016

CLINTON T. CONLEY
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA; clintonc@andrew.cmu.edu
ANDREW S. MARKS
Affiliation:
UCLA Mathematics, BOX 951555, Los Angeles, CA 90095-1555, USA; marks@math.ucla.edu
ROBIN D. TUCKER-DROB
Affiliation:
Texas A&M Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368, USA; rtuckerd@math.tamu.edu

Abstract

We generalize Brooks’ theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d\geqslant 3$ which contains no $(d+1)$ -cliques, then $G$ admits a ${\it\mu}$ -measurable $d$ -coloring with respect to any Borel probability measure ${\it\mu}$ on $X$ , and a Baire measurable $d$ -coloring with respect to any compatible Polish topology on $X$ . The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID $d$ -colorings of Cayley graphs of degree $d$ , except in two exceptional cases.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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