Related papers: Exact Chromatic Polynomials for Toroidal Chains of…

We calculate the continuous accumulation set ${\cal B}_q(p,\ell)$ of zeros of the chromatic polynomial $P(G^{(p,\ell)}_m,q)$ in the limit $m \to \infty$, on a family of graphs $G^{(p,\ell)}_m$ defined such that $G^{(p,\ell)}_m$ is obtained…

We calculate the chromatic polynomials $P$ for $n$-vertex strip graphs of the form $J(\prod_{\ell=1}^m H)I$, where $J$ and $I$ are various subgraphs on the left and right ends of the strip, whose bulk is comprised of $m$-fold repetitions of…

We present exact calculations of the zero-temperature partition function (chromatic polynomial) $P$ for the $q$-state Potts antiferromagnet on triangular lattice strips of arbitrarily great length $L_x$ vertices and of width $L_y=3$…

We prove that the chromatic polynomial $P_\mathbb{G}(q)$ of a finite graph $\mathbb{G}$ of maximal degree $\D$ is free of zeros for $\card q\ge C^*(\D)$ with $$ C^*(\D) = \min_{0<x<2^{1\over \D}-1} {(1+x)^{\D-1}\over x [2-(1+x)^\D]} $$ This…

We present exact calculations of chromatic polynomials for families of cyclic graphs consisting of linked polygons, where the polygons may be adjacent or separated by a given number of bonds. From these we calculate the (exponential of the)…

We report exact results concerning the zeros of the partition function of the Potts model in the complex $q$ plane, as a function of a temperature-like Boltzmann variable $v$, for the $m$'th iterate graphs $D_m$ of the Diamond Hierarchical…

We study the chromatic polynomial P_G(q) for m \times n square- and triangular-lattice strips of widths 2\leq m \leq 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the…

The chromatic polynomial is a well studied object in graph theory. There are many results and conjectures about the log-concavity of the chromatic polynomial and other polynomials related to it. The location of the roots of these…

We give algorithms for approximating the partition function of the ferromagnetic $q$-color Potts model on graphs of maximum degree $d$. Our primary contribution is a fully polynomial-time approximation scheme for $d$-regular graphs with an…

We study properties of the Potts model partition function $Z(H_m,q,v)$ on $m$'th iterates of Hanoi graphs, $H_m$, and use the results to draw inferences about the $m \to \infty$ limit that yields a self-similar Hanoi fractal, $H_\infty$. We…

We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(x)$ (in $\mathbb{C}$) lie inside the disc of radius $5.94 \Delta$ centered at $0$. This improves on the previously best known…

We prove that for any graph $G$ the (complex) zeros of its chromatic polynomial, $\chi_G(x)$, lie inside the disk centered at $0$ of radius $4.25 \Delta(G)$, where $\Delta(G)$ denotes the maximum degree of $G$. This improves on a recent…

Chromatic polynomials have been studied extensively, giving us results such as the Fundamental Reduction Theorem and closed formulas for the chromatic polynomials of common classes of graphs. Though, none of those extend to the context of…

A complete partition of a graph $G$ is a partition of the vertex set such that there is at least one edge between any two parts. The largest $r$ such that $G$ has a complete partition into $r$ parts, each of which is an independent set, is…

We consider proper colorings of planar graphs embedded in the annulus, such that vertices on one rim can take Q_s colors, while all remaining vertices can take Q colors. The corresponding chromatic polynomial is related to the partition…

We present transfer matrices for the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial) on cyclic and M\"obius strips of the square, triangular, and honeycomb lattices of…

Here we observe that list coloring in graph theory coincides with the zero-temperature antiferromagnetic Potts model with an external field. We give a list coloring polynomial that equals the partition function in this case. This is…

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal…

We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$. This immediately…

We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(z)$ (in $\mathbb{C}$) lie outside the disk of radius $5.02 \Delta$ centered at $0$. This improves on the previously best known…