20,899 research outputs found
Two-Page Book Embeddings of 4-Planar Graphs
Back in the Eighties, Heath showed that every 3-planar graph is
subhamiltonian and asked whether this result can be extended to a class of
graphs of degree greater than three. In this paper we affirmatively answer this
question for the class of 4-planar graphs. Our contribution consists of two
algorithms: The first one is limited to triconnected graphs, but runs in linear
time and uses existing methods for computing hamiltonian cycles in planar
graphs. The second one, which solves the general case of the problem, is a
quadratic-time algorithm based on the book-embedding viewpoint of the problem.Comment: 21 pages, 16 Figures. A shorter version is to appear at STACS 201
Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths
When can a plane graph with prescribed edge lengths and prescribed angles
(from among \}) be folded flat to lie in an
infinitesimally thin line, without crossings? This problem generalizes the
classic theory of single-vertex flat origami with prescribed mountain-valley
assignment, which corresponds to the case of a cycle graph. We characterize
such flat-foldable plane graphs by two obviously necessary but also sufficient
conditions, proving a conjecture made in 2001: the angles at each vertex should
sum to , and every face of the graph must itself be flat foldable.
This characterization leads to a linear-time algorithm for testing flat
foldability of plane graphs with prescribed edge lengths and angles, and a
polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure
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
Rainbow Hamilton cycles in random regular graphs
A rainbow subgraph of an edge-coloured graph has all edges of distinct
colours. A random d-regular graph with d even, and having edges coloured
randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with
probability tending to 1 as n tends to infinity, provided d is at least 8.Comment: 16 page
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