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Nowhere-harmonic colorings of graphs

Authors: Matthias Beck and Benjamin Braun
Journal: Proc. Amer. Math. Soc. 140 (2012), 47-63
MSC (2010): Primary 05C78; Secondary 05A15, 52B20, 52C35
Published electronically: May 9, 2011
MathSciNet review: 2833516
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Abstract: Proper vertex colorings of a graph are related to its boundary map $ \partial_1$, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, $ L=\partial_1 \partial_1^t$, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley's theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss several examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings.

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Additional Information

Matthias Beck
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132

Benjamin Braun
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

Keywords: Nowhere-harmonic coloring, chromatic polynomial, boundary map, graph Laplacian, inside-out polytope, hyperplane arrangement.
Received by editor(s): July 13, 2010
Received by editor(s) in revised form: November 2, 2010
Published electronically: May 9, 2011
Additional Notes: This research was partially supported by the NSF through grants DMS-0810105 (first author) and DMS-0758321 (second author). The authors would like to thank Tom Zaslavsky and the anonymous referees for their comments and suggestions.
Communicated by: Jim Haglund
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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