Relation to Statistical Physics (Glauber Dynamics)

The connectedness of the graph of vertex colourings has been given some attention by statisti-cal physicians when studying the Glauber dynamics of an anti-ferromagnetic Potts model at zero temperature.

Almost uniform samplingenables us to approximately count structures of exponential size in polynomial time [51]. Often the sampling is applied by simulating an appropriate Markov chain. A rapidly mixingMarkov chain is one that, in simple terms, converges to a very close approximation of the stationary distribution in polynomial time [25]. Such a Markov chain, which is used for sampling k-colourings of a graph, is known as Glauber dynamics.

The Potts model is a statistical model used to study the mechanics of the particles in a crys-talline lattice. Studying the interaction of spins of the particles in this model offers a theoretical basis for describing ferromagnetism and other phenomena related to the physics of solids. In the ferromagnetic case, same spins of neighbouring particles are caused by a form of reduction of the total energy of the system which in its turn is caused by the existence of neighbouring pairs of particles with the same spin. In the anti-ferromagnetic case, neighbouring particles are urged to have different spins. In both cases, the temperature of the system is a measure of the tension of different spins to appear. As the temperature gets lower, the energy of the system is reduced more than the existence of neighbouring pairs of particles with same/different spins. At zero temperature the described causal relation becomes even more evident in the anti-ferromagnetic case.

The zero temperature anti-ferromagnetic state(spin) Potts model can be modelled as a k-colouring of a graph G, where the graph is the crystalline lattice (the vertices are the particles) and colours represent the possible spins. Neighbouring particles have different spins under the specific conditions, thus neighbouring vertices have different colours. Thus, the rapidly mixing Glauber dynamics Markov chain of the above model, describes the transition states of the spins of the system. The ‘rapidly mixing’ part of this model and transition state system is the closest related to our research. One of the conditions for a Glauber dynamics Markov chain to be rapidly mixing, is that the graph model has to be k-mixing. Of course, in this case the graphs are of a

3.4. APPLICATIONS 37 very specific class (lattices), and the number of colours is large enough in order to guarantee the k-mixing property (See Theorem 2.2.1).

For some more details on Markov Chains in this context and mixing times of combinatorial objects, see Jerrum’s book [51].

Chapter 4

Recolouring Chordal and Chordal Bipartite Graphs

In this chapter, we introduce a class of k-colourable graphs, which we call k-colour-dense and we show that the reconfiguration graph R`(G) of vertex colourings of a k-colour-dense G on n vertices is connected, when ` ≥ 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(n²).

Recall that the reconfiguration graph of the k-colourings of a graph G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge in the reconfiguration graph if they differ in colour on just one vertex of G.

Apart from the fundamental problem of characterising the relationship between the complexity of reconfiguration problems and their original version, it is also of interest to find shortest paths between solutions. The diameter of the reconfiguration graph provides an upper bound. This is also related to the complexity of finding paths in the reconfiguration graph between given solutions since paths of polynomial length in the reconfiguration graph are certificates for the problem being in NP.

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For any graph G on n vertices, the diameter of Rk(G), the reconfiguration graph of k-colourings of G, has been shown to be O(n²), if k = 3 and R3(G) is connected [17]. Although there are cases where Rk(G) is not connected but contains components of super-polynomial diameter [13], there is no known example of a family of graphs for which R_k(G) is connected but does not have O(n²) diameter.

A good place to start when thinking about the above question is to consider graphs of bounded degeneracy. A graph G of degeneracy k is such that for every subgraph H ⊂ G, H has a vertex of degree k. It is well known that graphs of degeneracy k are (k + 1)-colourable. Bonsma and Cereceda [13] showed that if G is a graph of degeneracy k, then R_k+2(G), the reconfiguration graph of (k + 2)-colourings of G, is connected. In light of what is already known, we are naturally led to ask whether R_k+2(G) has quadratic diameter; indeed it is conjectured [13] that R_k+2(G) has cubic diameter, although this is modified to quadratic [15]. Our work includes an important class of k-degenerate graphs, namely (k + 1)-colourable chordal graphs, for which we show the conjecture to be true.

4.1 Preliminaries

In this section we give some basic terminology and notation in addition to what is defined in Section 1.1.2.

The disjoint union of two vertex-disjoint graphs G1 = (V1, E1) and G2 = (V2, E2), which we denote by G₁∪ G₂, is the graph with vertex set V₁∪ V₂and edge set E₁∪ E₂.

A maximal connected subgraph D of a graph is called a connected component (or just compo-nent) of G; we shall often abuse notation by denoting both the connected component and its vertex set by D. A separator of a graph G = (V, E) is a set S ⊂ V such that G − S has more connected components than G; if two vertices u and v that belong to the same connected component in G are in two different connected components of G − S, then we say that S separates u and v. We say that we identify two vertices u and v if we replace them by a new vertex adjacent to all neighbours of u and v.

4.2. SUFFICIENT CONDITIONS FOR QUADRATIC DIAMETER 41