239 research outputs found
Colourings of cubic graphs inducing isomorphic monochromatic subgraphs
A -bisection of a bridgeless cubic graph is a -colouring of its
vertex set such that the colour classes have the same cardinality and all
connected components in the two subgraphs induced by the colour classes
(monochromatic components in what follows) have order at most . Ban and
Linial conjectured that every bridgeless cubic graph admits a -bisection
except for the Petersen graph. A similar problem for the edge set of cubic
graphs has been studied: Wormald conjectured that every cubic graph with
has a -edge colouring such that the two
monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose
components are paths). Finally, Ando conjectured that every cubic graph admits
a bisection such that the two induced monochromatic subgraphs are isomorphic.
In this paper, we give a detailed insight into the conjectures of Ban-Linial
and Wormald and provide evidence of a strong relation of both of them with
Ando's conjecture. Furthermore, we also give computational and theoretical
evidence in their support. As a result, we pose some open problems stronger
than the above mentioned conjectures. Moreover, we prove Ban-Linial's
conjecture for cubic cycle permutation graphs.
As a by-product of studying -edge colourings of cubic graphs having linear
forests as monochromatic components, we also give a negative answer to a
problem posed by Jackson and Wormald about certain decompositions of cubic
graphs into linear forests.Comment: 33 pages; submitted for publicatio
Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking
We propose a new camera-based method of robot identification, tracking and
orientation estimation. The system utilises coloured lights mounted in a circle
around each robot to create unique colour sequences that are observed by a
camera. The number of robots that can be uniquely identified is limited by the
number of colours available, , the number of lights on each robot, , and
the number of consecutive lights the camera can see, . For a given set of
parameters, we would like to maximise the number of robots that we can use. We
model this as a combinatorial problem and show that it is equivalent to finding
the maximum number of disjoint -cycles in the de Bruijn graph
.
We provide several existence results that give the maximum number of cycles
in in various cases. For example, we give an optimal
solution when . Another construction yields many cycles in larger
de Bruijn graphs using cycles from smaller de Bruijn graphs: if
can be partitioned into -cycles, then
can be partitioned into -cycles for any divisor of
. The methods used are based on finite field algebra and the combinatorics
of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied
Mathematic
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
The Complexity of Counting Homomorphisms to Cactus Graphs Modulo 2
A homomorphism from a graph G to a graph H is a function from V(G) to V(H)
that preserves edges. Many combinatorial structures that arise in mathematics
and computer science can be represented naturally as graph homomorphisms and as
weighted sums of graph homomorphisms. In this paper, we study the complexity of
counting homomorphisms modulo 2. The complexity of modular counting was
introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who
famously introduced a problem for which counting modulo 7 is easy but counting
modulo 2 is intractable. Modular counting provides a rich setting in which to
study the structure of homomorphism problems. In this case, the structure of
the graph H has a big influence on the complexity of the problem. Thus, our
approach is graph-theoretic. We give a complete solution for the class of
cactus graphs, which are connected graphs in which every edge belongs to at
most one cycle. Cactus graphs arise in many applications such as the modelling
of wireless sensor networks and the comparison of genomes. We show that, for
some cactus graphs H, counting homomorphisms to H modulo 2 can be done in
polynomial time. For every other fixed cactus graph H, the problem is complete
for the complexity class parity-P which is a wide complexity class to which
every problem in the polynomial hierarchy can be reduced (using randomised
reductions). Determining which H lead to tractable problems can be done in
polynomial time. Our result builds upon the work of Faben and Jerrum, who gave
a dichotomy for the case in which H is a tree.Comment: minor change
Decomposing cubic graphs into isomorphic linear forests
A common problem in graph colouring seeks to decompose the edge set of a
given graph into few similar and simple subgraphs, under certain divisibility
conditions. In 1987 Wormald conjectured that the edges of every cubic graph on
vertices can be partitioned into two isomorphic linear forests. We prove
this conjecture for large connected cubic graphs. Our proof uses a wide range
of probabilistic tools in conjunction with intricate structural analysis, and
introduces a variety of local recolouring techniques.Comment: 49 pages, many figure
- …