4,021 research outputs found
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
Finite length spectra of random surfaces and their dependence on genus
The main goal of this article is to understand how the length spectrum of a
random surface depends on its genus. Here a random surface means a surface
obtained by randomly gluing together an even number of triangles carrying a
fixed metric.
Given suitable restrictions on the genus of the surface, we consider the
number of appearances of fixed finite sets of combinatorial types of curves. Of
any such set we determine the asymptotics of the probability distribution. It
turns out that these distributions are independent of the genus in an
appropriate sense.
As an application of our results we study the probability distribution of the
systole of random surfaces in a hyperbolic and a more general Riemannian
setting. In the hyperbolic setting we are able to determine the limit of the
probability distribution for the number of triangles tending to infinity and in
the Riemannian setting we derive bounds.Comment: 30 pages, 6 figure
Cycle Double Covers and Integer Flows
My research focuses on two famous problems in graph theory, namely the cycle double cover conjecture and the integer flows conjectures. This kind of problem is undoubtedly one of the major catalysts in the tremendous development of graph theory. It was observed by Tutte that the Four color problem can be formulated in terms of integer flows, as well as cycle covers. Since then, the topics of integer flows and cycle covers have always been in the main line of graph theory research. This dissertation provides several partial results on these two classes of problems
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