In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models.More specifically, we define a large class of graph polynomials C and show that if p ∈ C and there is a disk D centered at zero in the complex plane such that p(G) does not vanish on D for all bounded degree graphs G, then for each z in the interior of D there exists a deterministic polynomial-time approximation algorithm for evaluating p(G) at z. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures.Our work builds on a recent line of work initiated by Barvinok [2,3,4,5], which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.

A k-colouring of a graph G = (V, E) is a mapping c : V → {1, 2, . . . , k} such that c(u) = c(v) whenever uv is an edge. The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of k-colourable graphs, which we call k-colour-dense graphs. We show that for each k-colour-dense graph G, the reconfiguration graph of the ℓ-colourings of G is connected and has diameter O(|V | 2 ), for all ℓ ≥ k + 1. We show that this graph class contains the k-colourable chordal graphs and that it contains all chordal bipartite graphs when k = 2. Moreover, we prove that for each k ≥ 2 there is a k-colourable chordal graph G whose reconfiguration graph of the (k + 1)-colourings has diameter Θ(|V | 2 ).

The toughness of a (noncomplete) graph G is the minimum value of t for which there is a vertex cut A whose removal yields |A|/t components. Determining toughness is an NP‐hard problem for general input graphs. The toughness conjecture of Chvátal, which states that there exists a constant t such that every graph on at least three vertices with toughness at least t is hamiltonian, is still open for general graphs. We extend some known toughness results for split graphs to the more general class of 2K2‐free graphs, that is, graphs that do not contain two vertex‐disjoint edges as an induced subgraph. We prove that the problem of determining toughness is polynomially solvable and that Chvátal's toughness conjecture is true for 2K2‐free graphs.

ABSTRACT:We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree ( ) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on the number of colours such that Glauber dynamics is rapidly mixing if at least L colours are used. We give a closely-matching upper bound U on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random -regular graphs when at most U colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree , e.g. toroidal grids for = 4.