5,919 research outputs found
On the non-ergodicity of the Swendsen-Wang-Kotecky algorithm on the kagome lattice
We study the properties of the Wang-Swendsen-Kotecky cluster Monte Carlo
algorithm for simulating the 3-state kagome-lattice Potts antiferromagnet at
zero temperature. We prove that this algorithm is not ergodic for symmetric
subsets of the kagome lattice with fully periodic boundary conditions: given an
initial configuration, not all configurations are accessible via Monte Carlo
steps. The same conclusion holds for single-site dynamics.Comment: Latex2e. 22 pages. Contains 11 figures using pstricks package. Uses
iopart.sty. Final version accepted in journa
A reconfigurations analogue of Brooks’ theorem.
Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless
G is a complete graph or a cycle with an odd number of vertices, or
k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike.
We use this result to study the reconfiguration graph R k (G) of the k-colourings of G. The vertex set of R k (G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that
if k ≤ Δ(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter,
if k ≥ Δ(G) + 2, then R k (G) is connected and has diameter O(n 2).
We complete this structural classification by settling the missing case:
if k = Δ(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2).
We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R k (G) by settling the case k = Δ(G) + 1. The problem is
O(n 2) time solvable for k = 3,
PSPACE-complete for 4 ≤ k ≤ Δ(G),
O(n) time solvable for k = Δ(G) + 1,
O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes)
A Dichotomy Theorem for Circular Colouring Reconfiguration
The "reconfiguration problem" for circular colourings asks, given two
-colourings and of a graph , is it possible to transform
into by changing the colour of one vertex at a time such that every
intermediate mapping is a -colouring? We show that this problem can be
solved in polynomial time for and is PSPACE-complete for
. This generalizes a known dichotomy theorem for reconfiguring
classical graph colourings.Comment: 22 pages, 5 figure
Lattices of Graphical Gaussian Models with Symmetries
In order to make graphical Gaussian models a viable modelling tool when the
number of variables outgrows the number of observations, model classes which
place equality restrictions on concentrations or partial correlations have
previously been introduced in the literature. The models can be represented by
vertex and edge coloured graphs. The need for model selection methods makes it
imperative to understand the structure of model classes. We identify four model
classes that form complete lattices of models with respect to model inclusion,
which qualifies them for an Edwards-Havr\'anek model selection procedure. Two
classes turn out most suitable for a corresponding model search. We obtain an
explicit search algorithm for one of them and provide a model search example
for the other.Comment: 29 pages, 18 figures. Restructured Section 5, results unchanged;
added references in Section 6; amended example in Section 6.
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