Chromatic polynomials of random graphs

Abstract

Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse, or highly structured. Recent algorithmic advances (New J. Phys. 11:023001, 2009) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.

Characteristic Power Series of Graph Limits

“…For which other graph polynomials can the above type of limit process be carried out? By [10], chromatic polynomials of Erdős-Rényi random graphs appear to have a scaling limit, for example.…”

The structure of chromatic polynomials of planar triangulations and implications for chromatic zeros and asymptotic limiting quantities

“…Our results are of interest both from the point of view of mathematical graph theory and statistical physics and further show the fruitful connections between these fields. In future work, one could also investigate P(G pt , q) for arbitrary planar triangulations, including random ones [36].…”

Approximating the Chromatic Polynomial

“…If v i v j ∈ E(G), and σ vi = σ vj , then (1 − δ σv i σv j ) = 0 indicating an improper coloring -this assignment of σ vi 's is thus not included in the sum. In this manner, we may also interpret σ as a 'global microscopic state of an anti-ferromagnetic Potts model with the individual σ vi 's being local states or spin values' [30]. Thus, (1) can be used to count energy minimizing global states.…”

“…Unfortunately, computing P (G, x) for a general graph G is known to be #P -hard [16,22] and deciding whether or not a graph is k-colorable is N P -hard [11]. Polynomial-time algorithms have been found for certain subclasses of graphs, including chordal graphs [21] and graphs of bounded clique-width [12,20], and recent advances have made it feasible to study P (G, x) for graphs of up to thirty vertices [29,30]. Still, the best known algorithm for computing P (G, x) for an arbitrary graph G of order n has complexity O(2 n n O (1) ) [4] and the best current implementation is limited to 2|E(G)| + |V (G)| < 950 and |V (G)| < 65 [14,15].…”