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Binomial Inequalities for Chromatic, Flow, and Tension Polynomials

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Abstract

A famous and wide-open problem, going back to at least the early 1970s, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute such inequalities when a chromatic polynomial \(\chi _G(n)=\chi ^*_0\left( {\begin{array}{c}n+d\\ d\end{array}}\right) +\chi ^*_1\left( {\begin{array}{c}n+d-1\\ d\end{array}}\right) +\dots +\chi ^*_d\left( {\begin{array}{c}n\\ d\end{array}}\right) \) is written in terms of a binomial-coefficient basis. For example, we show that \(\chi ^*_j\le \chi ^*_{d-j}\), for \(0\le j\le d/2\). Similar results hold for flow and tension polynomials enumerating either modular or integral nowhere-zero flows/tensions of a graph. Our theorems follow from connections among chromatic, flow, tension, and order polynomials, as well as Ehrhart polynomials of lattice polytopes that admit unimodular triangulations. Our results use Ehrhart inequalities due to Athanasiadis and Stapledon and are related to recent work by Hersh–Swartz and Breuer–Dall, where inequalities similar to some of ours were derived using algebraic-combinatorial methods.

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Notes

  1. If we do not specify a basis when talking about the coefficients of a polynomial, we are thinking of the standard monomial basis.

  2. There is a subtlety here that differentiates \(\chi _G(n)\) from \(\phi _G(n)\), \(f_G(n)\), and \(t_G(z)\), and thus one needs to treat the accompanying generating functions with some care. Namely, \(\chi _G(n)\) has constant term 0, which is not true for \(\phi _G(n)\), \(f_G(n)\), and \(t_G(z)\). Note that we chose all of our generating functions to start with \(n=1\); the alternative choice of starting with \(n=0\) would result in a different definition of the binomial transform.

  3. The equality (7) of \(h^*_{\mathscr {O}}(z)\) and \(h^*_{ \mu ({\mathscr {O}}) } (z)\) can be also seen by noticing that the triangulations T and \(T_\mu \) are regular and therefore shellable, and they have the same h-polynomial. See, e.g., [11] why order polytopes are compressed, and therefore have regular unimodular triangulations, and also how these properties are preserved under the projection \(\mu \). We also note that projected order polytopes are examples of alcoved polytopes [17].

  4. An orientation is acyclic if it does not contain any coherently directed cycles.

  5. An orientation is totally cyclic if every edge lies in a coherently directed cycle.

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Acknowledgements

We thank Tristram Bogart, Tricia Hersh, Florian Kohl, Julián Pulido, Alan Stapledon, Tom Zaslavsky, and two anonymous referees for many useful conversations and comments. Also thanks to the organizers of ECCO 2016 (Escuela Colombiana de Combinatoria 2016 in Medellín, Colombia) where this research collaboration started, and to the Universidad de los Andes for their support.

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Authors and Affiliations

  1. Department of Mathematics, San Francisco State University, San Francisco, USA

    Matthias Beck

  2. Mathematisches Institut, Freie Universität Berlin, Berlin, Germany

    Matthias Beck

  3. Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia

    Emerson León

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A preliminary version of this paper [18] appeared as an extended abstract in the conference proceedings for FPSAC 2017 (Formal Power Series and Combinatorics at Queen Mary University of London), published in Séminaire Lotharingien Combinatoire.

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Beck, M., León, E. Binomial Inequalities for Chromatic, Flow, and Tension Polynomials. Discrete Comput Geom 66, 464–474 (2021). https://doi.org/10.1007/s00454-021-00314-3

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