1,452 research outputs found
Colouring random graphs and maximising local diversity
We study a variation of the graph colouring problem on random graphs of
finite average connectivity. Given the number of colours, we aim to maximise
the number of different colours at neighbouring vertices (i.e. one edge
distance) of any vertex. Two efficient algorithms, belief propagation and
Walksat are adapted to carry out this task. We present experimental results
based on two types of random graphs for different system sizes and identify the
critical value of the connectivity for the algorithms to find a perfect
solution. The problem and the suggested algorithms have practical relevance
since various applications, such as distributed storage, can be mapped onto
this problem.Comment: 10 pages, 10 figure
Minimizing Unsatisfaction in Colourful Neighbourhoods
Colouring sparse graphs under various restrictions is a theoretical problem
of significant practical relevance. Here we consider the problem of maximizing
the number of different colours available at the nodes and their
neighbourhoods, given a predetermined number of colours. In the analytical
framework of a tree approximation, carried out at both zero and finite
temperatures, solutions obtained by population dynamics give rise to estimates
of the threshold connectivity for the incomplete to complete transition, which
are consistent with those of existing algorithms. The nature of the transition
as well as the validity of the tree approximation are investigated.Comment: 28 pages, 12 figures, substantially revised with additional
explanatio
Colouring quadrangulations of projective spaces
A graph embedded in a surface with all faces of size 4 is known as a
quadrangulation. We extend the definition of quadrangulation to higher
dimensions, and prove that any graph G which embeds as a quadrangulation in the
real projective space P^n has chromatic number n+2 or higher, unless G is
bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996),
219-227]. The family of quadrangulations of projective spaces includes all
complete graphs, all Mycielski graphs, and certain graphs homomorphic to
Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser
theorem
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
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