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Chromatic Polynomial Evaluation Spectra

Rafael Miyazaki, Cosmin Pohoata, Michael Zheng

TL;DR

This work proves that, for any fixed q not in {0,1,2}, the number of distinct chromatic polynomial values P_G(q) realized by n-vertex graphs grows exponentially with n, and provides explicit bounds in several regimes. The authors synthesize negative-evaluation reciprocity, a boundary-vector framework, and a ping–pong argument using two simple graph operations to generate exponentially many distinct evaluations on planar graphs; they then extend the results to non-integer positive q via a clique-join trick and to q>2 via attainable-vector techniques. For q>2 and q∈(0,2)ackslash{1}, they obtain planar bounds of the form |S_n^{pl}(q)| ≥ √F_{n−2} and 2^{Ω(n)}, respectively, and for q<0 they obtain |S_n^{pl}(q)| ≥ 2^{(n−2)/2}. The results deepen our understanding of chromatic polynomial spectra and suggest similar phenomena for broader Tutte-type invariants.

Abstract

Around 10 years ago, Agol and Krushkal showed that the number of chromatic polynomials $P_{G}$ arising from graphs $G$ on $n$ vertices grows exponentially with $n$, by establishing that the (dual) flow polynomial $F_{G}\left(\frac{3+\sqrt{5}}{2}\right)$ already takes on exponentially many values, if one varies $G$ over all planar cubic graphs $G$ on $n$ vertices. We show, more generally, that the size of the set $\{P_G(q): |V(G)|=n\}$ is exponential in $n$, for every fixed real number $q \neq 0,1,2$. In fact, our approach can also be pushed to show that $P_{G}(q)$ already takes on exponentially many values, if we only vary $G$ over all planar graphs on $n$ vertices. The case $q=3$ confirms a conjecture of Agol, which was initially motivated by the $\mathsf{NP}$-completeness of planar $3$-colorability.

Chromatic Polynomial Evaluation Spectra

TL;DR

This work proves that, for any fixed q not in {0,1,2}, the number of distinct chromatic polynomial values P_G(q) realized by n-vertex graphs grows exponentially with n, and provides explicit bounds in several regimes. The authors synthesize negative-evaluation reciprocity, a boundary-vector framework, and a ping–pong argument using two simple graph operations to generate exponentially many distinct evaluations on planar graphs; they then extend the results to non-integer positive q via a clique-join trick and to q>2 via attainable-vector techniques. For q>2 and q∈(0,2)ackslash{1}, they obtain planar bounds of the form |S_n^{pl}(q)| ≥ √F_{n−2} and 2^{Ω(n)}, respectively, and for q<0 they obtain |S_n^{pl}(q)| ≥ 2^{(n−2)/2}. The results deepen our understanding of chromatic polynomial spectra and suggest similar phenomena for broader Tutte-type invariants.

Abstract

Around 10 years ago, Agol and Krushkal showed that the number of chromatic polynomials arising from graphs on vertices grows exponentially with , by establishing that the (dual) flow polynomial already takes on exponentially many values, if one varies over all planar cubic graphs on vertices. We show, more generally, that the size of the set is exponential in , for every fixed real number . In fact, our approach can also be pushed to show that already takes on exponentially many values, if we only vary over all planar graphs on vertices. The case confirms a conjecture of Agol, which was initially motivated by the -completeness of planar -colorability.
Paper Structure (15 sections, 19 theorems, 68 equations)

This paper contains 15 sections, 19 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Negative evaluations and chromatic reciprocity
  3. A sign-normalization and what it counts at negative integers
  4. Positivity and an additive deletion--contraction identity
  5. Feasible vectors and a square-root lemma
  6. Two basic operations and their matrices
  7. Ping--pong on ratios and exponential growth
  8. Non-integer positive evaluations via clique joins
  9. Join with a clique shifts the evaluation point
  10. Exponential growth for every non-integer q > 0
  11. The planar evaluation spectrum for q > 2
  12. Attainable vectors
  13. Ping--pong on ratios and exponential growth
  14. A planar ping--pong for q in (0,2) - 1
  15. Concluding remarks

Key Result

Theorem 1.1

For every real number $q\notin\{0,1,2\}$, the evaluation spectrum $|\mathsf{S}_n^{\mathop{\mathrm{\mathrm{pl}}}\nolimits}(q)|$ grows exponentially in $n$. Moreover, we can show the following uniform lower bounds: where $(F_n)_n$ is the Fibonacci sequence and $\varphi = (1+\sqrt{5})/2$ is the golden ratio.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: Chromatic reciprocity Stanley1973
  • Remark 2.2
  • Lemma 2.3: Positivity for $\lambda>0$
  • proof
  • Lemma 2.4: Additive deletion--contraction for $Z$
  • proof
  • Definition 2.5: Feasible vectors
  • Lemma 2.6: Square-root lemma
  • ...and 37 more