162 research outputs found
Enumerating the edge-colourings and total colourings of a regular graph
In this paper, we are interested in computing the number of edge colourings and total colourings of a graph. We prove that the maximum number of -edge-colourings of a -regular graph on vertices is . Our proof is constructible and leads to a branching algorithm enumerating all the -edge-colourings of a -regular graph using a time and polynomial space. In particular, we obtain a algorithm on time and polynomial space to enumerate all the -edge colourings of a cubic graph, improving the running time of of the algorithm due to Golovach et al.~\cite{GKC10}. We also show that the number of -total-colourings of a connected cubic graph is at most . Again, our proof yields a branching algorithm to enumerate all the -total-colourings of a connected cubic graph
Dimer and fermionic formulations of a class of colouring problems
We show that the number Z of q-edge-colourings of a simple regular graph of
degree q is deducible from functions describing dimers on the same graph, viz.
the dimer generating function or equivalently the set of connected dimer
correlation functions. Using this relationship to the dimer problem, we derive
fermionic representations for Z in terms of Grassmann integrals with quartic
actions. Expressions are given for planar graphs and for nonplanar graphs
embeddable (without edge crossings) on a torus. We discuss exact numerical
evaluations of the Grassmann integrals using an algorithm by Creutz, and
present an application to the 4-edge-colouring problem on toroidal square
lattices, comparing the results to numerical transfer matrix calculations and a
previous Bethe ansatz study. We also show that for the square, honeycomb, 3-12,
and one-dimensional lattice, known exact results for the asymptotic scaling of
Z with the number of vertices can be expressed in a unified way as different
values of one and the same function.Comment: 16 pages, 2 figures, 2 tables. v2: corrected an inconsistency in the
notatio
A connection between the ice-type model of Linus Pauling and the three-color problem
The ice-type model proposed by Linus Pauling to explain its entropy at low temperatures is here approached in a didactic way. We first present a theoretically estimated low-temperature entropy and compare it with numerical results. Then, we consider the mapping between this model and the three-colour problem, i.e. colouring a regular graph with coordination equal to 4 (a twodimensional lattice) with three colours, for which we apply the transfer-matrix method to calculate all allowed configurations for two-dimensional square lattices of N oxygen atoms ranging from 4 to 225. Finally, from a linear regression of the transfer matrix results, we obtain an estimate for the case N → ∞ which is compared with the exact solution by Lieb
Exact Ramsey numbers of odd cycles via nonlinear optimisation
For a graph G, the k-colour Ramsey number R k(G) is the least integer N such that every k-colouring of the edges of the complete graph K N contains a monochromatic copy of G. Let C n denote the cycle on n vertices. We show that for fixed k≥2 and n odd and sufficiently large, R k(C n)=2 k−1(n−1)+1. This resolves a conjecture of Bondy and Erdős for large n. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation. This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal k-colourings for this problem and perfect matchings in the k-dimensional hypercube Q k
Fast Algorithms for Join Operations on Tree Decompositions
Treewidth is a measure of how tree-like a graph is. It has many important
algorithmic applications because many NP-hard problems on general graphs become
tractable when restricted to graphs of bounded treewidth. Algorithms for
problems on graphs of bounded treewidth mostly are dynamic programming
algorithms using the structure of a tree decomposition of the graph. The
bottleneck in the worst-case run time of these algorithms often is the
computations for the so called join nodes in the associated nice tree
decomposition.
In this paper, we review two different approaches that have appeared in the
literature about computations for the join nodes: one using fast zeta and
M\"obius transforms and one using fast Fourier transforms. We combine these
approaches to obtain new, faster algorithms for a broad class of vertex subset
problems known as the [\sigma,\rho]-domination problems. Our main result is
that we show how to solve [\sigma,\rho]-domination problems in arithmetic operations. Here, t is the treewidth, s is the
(fixed) number of states required to represent partial solutions of the
specific [\sigma,\rho]-domination problem, and n is the number of vertices in
the graph. This reduces the polynomial factors involved compared to the
previously best time bound (van Rooij, Bodlaender, Rossmanith, ESA 2009) of arithmetic operations. In particular, this removes
the dependence of the degree of the polynomial on the fixed number of
states~.Comment: An earlier version appeared in "Treewidth, Kernels, and Algorithms.
Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday"
LNCS 1216
Towards derandomising Markov chain Monte Carlo
We present a new framework to derandomise certain Markov chain Monte Carlo
(MCMC) algorithms.
As in MCMC, we first reduce counting problems to sampling from a sequence of
marginal distributions.
For the latter task,
we introduce a method called coupling towards the past that can, in
logarithmic time,
evaluate one or a constant number of variables from a stationary Markov chain
state.
Since there are at most logarithmic random choices, this leads to very simple
derandomisation.
We provide two applications of this framework, namely efficient deterministic
approximate counting algorithms for hypergraph independent sets and hypergraph
colourings,
under local lemma type conditions matching, up to lower order factors, their
state-of-the-art randomised counterparts.Comment: 57 page
Properties of random graphs
The thesis describes new results for several problems in random graph theory.
The first problem relates to the uniform random graph model in
the supercritical phase; i.e. a graph, uniformly distributed, on vertices
and edges for satisfying
and . The property studied is the length of the
longest cycle in the graph. We give a new upper bound, which holds
asymptotically almost surely, on this length.
As part of our proof we establish a result about the heaviest cycle in a certain
randomly-edge-weighted nearly-3-regular graph, which may be of independent interest.
Our second result is a new contiguity result for a random -regular
graph. Let be a function that is linear in .
A -irregular graph is a graph which is -regular except for
vertices of
degree . A -edge matching in a graph is a set of independent edges.
In this thesis we prove the new result that a random
-irregular graph plus a random -edge matching is contiguous to a random
-regular graph, in the sense that
in the two spaces,
the same events have probability approaching 1 as .
This allows one to deduce properties, such as colourability,
of the random irregular graph from
the corresponding properties of the random regular one. The proof
applies the small subgraph conditioning method to the number of -edge matchings
in a random -regular graph.
The third problem is about the 3-colourability of
a random 5-regular graph. Call a colouring balanced
if the number of vertices of each colour
is equal, and locally rainbow if every vertex is adjacent to vertices
of all the other
colours. Using the small subgraph conditioning method, we give a
condition on the variance of the number of locally rainbow balanced 3-colourings which, if
satisfied, establishes that the chromatic number of the random 5-regular graph is
asymptotically almost surely equal to 3.
We also describe related work which provides evidence that the condition is
likely to be true.
The fourth problem is about the chromatic number of a random -regular
graph for fixed .
Achlioptas and Moore recently announced a proof that a random -regular
graph asymptotically almost surely has chromatic number , , or ,
where is the smallest integer satisfying . In
this thesis we prove that, asymptotically almost surely, it is not ,
provided a certain second moment condition holds.
The proof applies the small subgraph conditioning method to
the number of balanced -colourings, where a colouring is balanced
if the number of vertices of each colour is equal.
We also give evidence that suggests that the required
second moment condition is true
Tube algebras, excitations statistics and compactification in gauge models of topological phases
We consider lattice Hamiltonian realizations of (+1)-dimensional
Dijkgraaf-Witten theory. In (2+1)d, it is well-known that the Hamiltonian
yields point-like excitations classified by irreducible representations of the
twisted quantum double. This can be confirmed using a tube algebra approach. In
this paper, we propose a generalization of this strategy that is valid in any
dimensions. We then apply the tube algebra approach to derive the algebraic
structure of loop-like excitations in (3+1)d, namely the twisted quantum
triple. The irreducible representations of the twisted quantum triple algebra
correspond to the simple loop-like excitations of the model. Similarly to its
(2+1)d counterpart, the twisted quantum triple comes equipped with a compatible
comultiplication map and an -matrix that encode the fusion and the braiding
statistics of the loop-like excitations, respectively. Moreover, we explain
using the language of loop-groupoids how a model defined on a manifold that is
-times compactified can be expressed in terms of another model in -lower
dimensions. This can in turn be used to recast higher-dimensional tube algebras
in terms of lower dimensional analogues.Comment: 71 page
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