We exhibit infinite families of planar triangulations with real chromatic roots arbitrarily close to 4, thus resolving a long-standing conjecture in the affirmative.

Planar Triangulations with Real Chromatic Roots Arbitrarily Close to 4

We exhibit infinite families of planar graphs with real chromatic roots arbitrarily close to 4, thus resolving a long-standing conjecture in the affirmative.

Planar triangulations with real chromatic roots arbitrarily close to four

In this paper, we present a new algorithm for computing the chromatic polynomial of a general graph G. Our method is based on the addition of edges and contraction of non-edges of G, the base case of the recursion being chordal graphs. The set of edges to be considered is taken from a triangulation of G. To achieve our goal, we use the properties of triangulations and clique-trees with respect to the previous operations, and guide our algorithm to efficiently divide the original problem.

Furthermore, we give some lower bounds of the general complexity of our method, and provide experimental results for several families of graphs.

Computation of chromatic polynomials using triangulations and clique trees

Chromatic polynomials are important objects in graph theory and statistical physics, but as a result of computational difficulties, their study is limited to graphs that are small, highly structured, or very sparse. We have devised and implemented two algorithms that approximate the coefficients of the chromatic polynomial $P(G,x)$, where $P(G,k)$ is the number of proper $k$-colorings of a graph $G$ for $k\in\mathbb{N}$. Our algorithm is based on a method of Knuth that estimates the order of a search tree. We compare our results to the true chromatic polynomial in several known cases, and compare our error with previous approximation algorithms.

Approximating the Chromatic Polynomial.

A class of graphs called generalized ladder graphs is defined. A sufficient condition for pairs of these graphs to be chromatically equivalent is proven. In addition a formula for the chromatic polynomial of a graph of this type is proven. Finally, the chromatic polynomials of special cases of these graphs are explicitly computed.

Chromatic Equivalence of Generalized Ladder Graphs.

The computation of chromatic polynomials

Using Thresholds to Compute Chromatic Polynomials.

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The computation of chromatic polynomials

Using Thresholds to Compute Chromatic Polynomials.