I show that there exist universal constants $C(r) < \infty$ such that, for
all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex)
of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore,
$C(r) \le 7.963906... r$. This result is a corollary of a more general result
on the zeros of the Potts-model partition function $Z_G(q, {v_e})$ in the
complex antiferromagnetic regime $|1 + v_e| \le 1$. The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of $Z_G(q,
{v_e})$ to a polymer gas, followed by verification of the
Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model
partition function. I also show that, for all loopless graphs $G$ of
second-largest degree $\le r$, the zeros of $P_G(q)$ lie in the disc $|q| <
C(r) + 1$. Along the way, I give a simple proof of a generalized (multivariate)
Brown-Colbourn conjecture on the zeros of the reliability polynomial for the
special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs
  of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of
  Proposition 4.1, and adds related discussion. To appear in Combinatorics,
  Probability & Computin

Sokal, Alan D.

arXiv.org e-Print Archive

CiteSeerX

English

1=d-expansions for the free energy of lattice animal models of a self-interacting branched polymer.

A bibliography on chromatic polynomials.

A contribution to the theory of chromatic polynomials.

A determinantal formula for the number of ways of coloring a map.

A general Lee{Yang theorem for one-component and multicomponent ferromagnets.

A logical expansion in mathematics.

A new proof of the existence and nontriviality of the '4 2 and '4 3 quantum  theories.

A ring in graph theory.

A set of topological invariants for graphs.

A short course on cluster expansions.

A symmetric function generalization of the chromatic polynomial of a graph.

A theorem on the exact solution of a spin system with a  magnetic eld.

A two-dimensional isotropic quantum antiferromagnet with unique disordered ground state.

A zero-free interval for chromatic polynomials of graphs.

A zero-free interval for chromatic polynomials.

Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem.

Algebraic Graph Theory, 2nd edn,

An alternate constructive approach to the '4 3 quantum  theory, and a possible destructive approach to '4 4.

An introduction to chromatic polynomials.

An update on the four-color theorem.

Asymptotic limits and zeros of chromatic polynomials and ground state entropy of Potts antiferromagnets.

Beitrag zum Verst¨ andnis der magnetischen Erscheinungen in festen K¨ orpern.

Beitrag zur Theorie des Ferromagnetismus.

Cell growth problems.

Chromatic polynomials and order ideals of monomials.

Chromatic polynomials for families of strip graphs and their asymptotic limits. Physica A 252 505{546; cond-mat/9712148 at xxx.lanl.gov.

Chromatic polynomials for J( Q H)I strip graphs and their asymptotic limits. Physica A 259 367{387; cond-mat/9807106 at xxx.lanl.gov.

Chromatic polynomials of large triangular lattices.

Chromatic polynomials, polygon trees, and outerplanar graphs.

Chromatic polynomials.

Chromatic roots are dense in the whole complex plane.

Chromatic roots of families of graphs.

Chromatic roots: some observations and conjectures.

Cluster expansion for abstract polymer models.

Combinatorial Enumeration,

Combinatorics and Commutative Algebra, 2nd edn, Birkh¨ auser,

Complexity: Knots, Colourings, and Counting,

Con studies of the Potts models.

Convergence of fugacity expansions for classical systems.

Convexity in the Theory of Lattice Gases,

Correlations in Ising ferromagnets, II: External magnetic

Coulomb systems at low density: A review.

Decay of correlations for in range interactions in unbounded spin systems.

End graph eects on chromatic polynomials for strip graphs of lattices and their asymptotic limits. Physica A 259 349{366; cond-mat/9807107 at xxx.lanl.gov.

Entropy for translation-invariant random-cluster measures.

Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole,

Ernst Ising (obituary).

Ernst Ising: physicist and teacher.

Estimates of semi-invariants for the Ising model at low temperatures.

Every planar map is four colorable: Part 1, Discharging.

Every planar map is four colorable: Part 2, Reducibility.

Every planar map is four colorable.

Families of graphs with chromatic zeros lying on circles.

Families of graphs with Wr(fGg;q) functions that are nonanalytic at 1=q =0 .Phys.

Finite-size scaling for Potts models.

Gauge Theories as a Problem of Constructive Quantum Field Theory and

Generalization of the Fortuin{Kasteleyn{Swendsen{ Wang representation and Monte Carlo algorithm.

Gibbs Measures and Phase Transitions, De Gruyter,

Graph colorings and related symmetric functions: ideas and applications: A description of results, interesting applications, and notable open problems.

Graph reducibility.

Graphs and series-parallel networks.

Ground state degeneracy of Potts antiferromagnets: Cases with noncompact W boundaries having multiple points at 1=q =0

Ground state degeneracy of Potts antiferromagnets: Homeomorphic classes with noncompact W boundaries. Physica A 265 186{223; condmat/9811410 at xxx.lanl.gov.

Ground state entropy of Potts antiferromagnets on homeomorphic families of strip graphs.

Ground state entropy of the Potts antiferromagnet on cyclic strip graphs.

History of the Lenz{Ising model.

Interfaces in the Potts model, I: Pirogov{Sinai theory of the Fortuin{Kasteleyn representation.

Is the four-color conjecture almost false?

Lattice animals: Rigorous results and wild guesses.

Lectures on Polytopes,

Limits of chromatic zeros of some families of maps.

Location of zeros of chromatic and related polynomials of graphs.

Lower bounds and series for the ground state entropy of the Potts antiferromagnet on Archimedean lattices and their duals.

Mayer expansions and the Hamilton{Jacobi equation, II: Fermions, dimensional reduction formulas.

Mayer expansions and the Hamilton{Jacobi equation.

On a general class of graph polynomials.

On chromatic polynomials and the golden ratio.

On the coloring of graphs.

On the foundations of combinatorial theory, I: Theory of M¨ obius functions.

On the number of trees in Zd.

On the random-cluster model, I: Introduction and relation to other models.

On the roots of chromatic polynomials.

Perturbation methods of the theory of Gibbsian

Phase transitions in lattice systems with random local properties.

Phases coexistence and surface tensions for the Potts model.

Potts model of magnetism (invited).

Probability Theory: Independence, Interchangeability, Martingales, 3rd edn,

q colourings of the triangular lattice.

Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory,

Recursive families of graphs.

Relationships between growth constants for animals and trees.

Rigorous results for branched polymer models with excluded volume.

Roots of the reliability polynomial.

Self-avoiding walks and trees in spread-out lattices.

Some cluster size and percolation problems.

Some generalized order-disorder transformations.

Statistical theory of equations of state and phase transitions, I: Theory of condensation.

Statistical theory of equations of state and phase transitions, II: Lattice gas and Ising model.

Statistics of two-dimensional lattices with four components.

Steiner trees, partial 2-trees, and minimum IFI networks.

Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model
  Partition Functions