425 research outputs found
Hyperbolic triangular buildings without periodic planes of genus two
We study surface subgroups of groups acting simply transitively on vertex
sets of certain hyperbolic triangular buildings. The study is motivated by
Gromov's famous surface subgroup question: Does every one-ended hyperbolic
group contain a subgroup which is isomorphic to the fundamental group of a
closed surface of genus at least 2? Here we consider surface subgroups of the
23 torsion free groups acting simply transitively on the vertices of hyperbolic
triangular buildings of the smallest non-trivial thickness. These groups gave
the first examples of cocompact lattices acting simply transitively on vertices
of hyperbolic triangular Kac-Moody buildings that are not right-angled. With
the help of computer searches we show, that in most of the cases there are no
periodic apartments invariant under the action of a genus two surface. The
existence of such an action would imply the existence of a surface subgroup,
but it is not known, whether the existence of a surface subgroup implies the
existence of a periodic apartment. These groups are the first candidates for
groups that have no surface subgroups arising from periodic apartments
The Ramsey Number for 3-Uniform Tight Hypergraph Cycles
Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .â.â., vn and edges v1v2v3, v2v3v4, .â.â., vnâ1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every redâblue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl
Strong edge-colouring of sparse planar graphs
A strong edge-colouring of a graph is a proper edge-colouring where each
colour class induces a matching. It is known that every planar graph with
maximum degree has a strong edge-colouring with at most
colours. We show that colours suffice if the graph has girth 6, and
colours suffice if or the girth is at least 5. In the
last part of the paper, we raise some questions related to a long-standing
conjecture of Vizing on proper edge-colouring of planar graphs
Hamiltonian cycles in maximal planar graphs and planar triangulations
In this thesis we study planar graphs, in particular, maximal planar graphs and general planar triangulations. In Chapter 1 we present the terminology and notations that will be used throughout the thesis and review some elementary results on graphs that we shall need. In Chapter 2 we study the fundamentals of planarity, since it is the cornerstone of this thesis. We begin with the famous Euler's Formula which will be used in many of our results. Then we discuss another famous theorem in graph theory, the Four Colour Theorem. Lastly, we discuss Kuratowski's Theorem, which gives a characterization of planar graphs. In Chapter 3 we discuss general properties of a maximal planar graph, G particularly concerning connectivity. First we discuss maximal planar graphs with minimum degree i, for i = 3; 4; 5, and the subgraph induced by the vertices of G with the same degree. Finally we discuss the connectivity of G, a maximal planar graph with minimum degree i. Chapter 4 will be devoted to Hamiltonian cycles in maximal planar graphs. We discuss the existence of Hamiltonian cycles in maximal planar graphs. Whitney proved that any maximal planar graph without a separating triangle is Hamiltonian, where a separating triangle is a triangle such that its removal disconnects the graph. Chen then extended Whitney's results and allowed for one separating triangle and showed that the graph is still Hamiltonian. Helden also extended Chen's result and allowed for two separating triangles and showed that the graph is still Hamiltonian. G. Helden and O. Vieten went further and allowed for three separating triangles and showed that the graph is still Hamiltonian. In the second section we discuss the question by Hakimi and Schmeichel: what is the number of cycles of length p that a maximal planar graph on n vertices could have in terms of n? Then in the last section we discuss the question by Hakimi, Schmeichel and Thomassen: what is the minimum number of Hamiltonian cycles that a maximal planar graph on n vertices could have, in terms of n? In Chapter 5, we look at general planar triangulations. Note that every maximal planar graph on n â„ 3 vertices is a planar triangulation. In the first section we discuss general properties of planar triangulations and then end with Hamiltonian cycles in planar triangulations
Generation and Properties of Snarks
For many of the unsolved problems concerning cycles and matchings in graphs
it is known that it is sufficient to prove them for \emph{snarks}, the class of
nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part
of this paper we present a new algorithm for generating all non-isomorphic
snarks of a given order. Our implementation of the new algorithm is 14 times
faster than previous programs for generating snarks, and 29 times faster for
generating weak snarks. Using this program we have generated all non-isomorphic
snarks on vertices. Previously lists up to vertices have been
published. In the second part of the paper we analyze the sets of generated
snarks with respect to a number of properties and conjectures. We find that
some of the strongest versions of the cycle double cover conjecture hold for
all snarks of these orders, as does Jaeger's Petersen colouring conjecture,
which in turn implies that Fulkerson's conjecture has no small counterexamples.
In contrast to these positive results we also find counterexamples to eight
previously published conjectures concerning cycle coverings and the general
cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated
and typos corrected. This version differs from the published one in that the
Arxiv-version has data about the automorphisms of snarks; Journal of
Combinatorial Theory. Series B. 201
Approximately coloring graphs without long induced paths
It is an open problem whether the 3-coloring problem can be solved in
polynomial time in the class of graphs that do not contain an induced path on
vertices, for fixed . We propose an algorithm that, given a 3-colorable
graph without an induced path on vertices, computes a coloring with
many colors. If the input graph is
triangle-free, we only need many
colors. The running time of our algorithm is if the input
graph has vertices and edges
Mixing graph colourings
This thesis investigates some problems related to graph colouring, or, more precisely, graph re-colouring. Informally, the basic question addressed can be phrased as follows. Suppose one is given a graph G whose vertices can be properly k-coloured, for some k â„ 2. Is it possible to transform any k-colouring of G into any other by recolouring vertices of G one at a time, making sure a proper k-colouring of G is always maintained? If the answer is in the affirmative, G is said to be k-mixing. The related problem of deciding whether, given two k-colourings of G, it is possible to transform one into the other by recolouring vertices one at a time, always maintaining a proper k-colouring of G, is also considered.
These questions can be considered as having a bearing on certain mathematical and âreal-worldâ problems. In particular, being able to recolour any colouring of a given graph to any other colouring is a necessary pre-requisite for the method of sampling colourings known as Glauber dynamics. The results presented in this thesis may also find application in the context of frequency reassignment: given that the problem of assigning radio frequencies in a wireless communications network is often modelled as a graph colouring problem, the task of re-assigning frequencies in such a network can be thought of as a graph recolouring problem.
Throughout the thesis, the emphasis is on the algorithmic aspects and the computational complexity of the questions described above. In other words, how easily, in terms of computational resources used, can they be answered? Strong results are obtained for the k = 3 case of the first question, where a characterisation theorem for 3-mixing graphs is given. For the second question, a dichotomy theorem for the complexity of the problem is proved: the problem is solvable in polynomial time for k †3 and PSPACE-complete for k ℠4. In addition, the possible length of a shortest sequence of recolourings between two colourings is investigated, and an interesting connection between the tractability of the problem and its underlying structure is established. Some variants of the above problems are also explored
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