Enumerating colorings, tensions and flows in cell complexes

  • Authors:

  • Matthias Beck;Felix Breuer;Logan Godkin;Jeremy L. Martin

  • Affiliations:

  • Department of Mathematics, San Francisco State University, 1600 Holloway Ave, San Francisco, CA 94132, USA;Department of Mathematics, San Francisco State University, 1600 Holloway Ave, San Francisco, CA 94132, USA;Department of Mathematics, University of Kansas, Lawrence, KS 66047, USA;Department of Mathematics, University of Kansas, Lawrence, KS 66047, USA

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2014

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Abstract

We study quasipolynomials enumerating proper colorings, nowhere-zero tensions, and nowhere-zero flows in an arbitrary CW-complex X, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may be either modular (with values in Z/kZ for some k) or integral (with values in {-k+1,...,k-1}). We obtain deletion-contraction recurrences and closed formulas for the chromatic, tension and flow quasipolynomials, assuming certain unimodularity conditions. We use geometric methods, specifically Ehrhart theory and inside-out polytopes, to obtain reciprocity theorems for all of the aforementioned quasipolynomials, giving combinatorial interpretations of their values at negative integers as well as formulas for the numbers of acyclic and totally cyclic orientations of X.