6,070 research outputs found
On the genera of polyhedral embeddings of cubic graph
In this article we present theoretical and computational results on the
existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs.
We also describe an efficient algorithm to compute all polyhedral embeddings of
a given cubic graph and constructions for cubic graphs with some special
properties of their polyhedral embeddings. Some key results are that even cubic
graphs with a polyhedral embedding on the torus can also have polyhedral
embeddings in arbitrarily high genus, in fact in a genus {\em close} to the
theoretical maximum for that number of vertices, and that there is no bound on
the number of genera in which a cubic graph can have a polyhedral embedding.
While these results suggest a large variety of polyhedral embeddings,
computations for up to 28 vertices suggest that by far most of the cubic graphs
do not have a polyhedral embedding in any genus and that the ratio of these
graphs is increasing with the number of vertices.Comment: The C-program implementing the algorithm described in this article
can be obtained from any of the author
Embedding Digraphs on Orientable Surfaces
AbstractWe consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of compatible Euler tours or orthogonal A-trails by Andersen and by Bouchet. This prior work has mostly been limited to embeddings of Eulerian digraphs on predetermined surfaces and to digraphs with underlying graphs of maximum degree at most 4. In this paper, a foundation is laid for the study of all Eulerian digraph embeddings. Results are proved which are analogous to those fundamental to the theory of undirected graph embeddings, such as Duke's theorem [5], and an infinite family of digraphs which demonstrates that the genus range for an embeddable digraph can be any nonnegative integer given. We show that it is possible to have genus range equal to one, with arbitrarily large minimum genus, unlike in the undirected case. The difference between the minimum genera of a digraph and its underlying graph is considered, as is the difference between the maximum genera. We say that a digraph is upper-embeddable if it can be embedded with two or three regions and prove that every regular tournament is upper-embeddable
One-sided curvature estimates for H-disks
In this paper we prove an extrinsic one-sided curvature estimate for disks
embedded in with constant mean curvature which is independent of
the value of the constant mean curvature. We apply this extrinsic one-sided
curvature estimate in [24] to prove to prove a weak chord arc type result for
these disks. In Section 4 we apply this weak chord arc result to obtain an
intrinsic version of the one-sided curvature estimate for disks embedded in
with constant mean curvature. In a natural sense, these
one-sided curvature estimates generalize respectively, the extrinsic and
intrinsic one-sided curvature estimates for minimal disks embedded in
given by Colding and Minicozzi in Theorem 0.2 of [8] and in
Corollary 0.8 of [9].Comment: Minor corrections. References updated. Format change
- …