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 | References | Similar Articles | Additional Information

Abstract: Proper vertex colorings of a graph are related to its boundary map , also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, , 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

Email:
beck@math.sfsu.edu

**Benjamin Braun**

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

Email:
benjamin.braun@uky.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-10879-7

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.