11,739 research outputs found
Coloring decompositions of complete geometric graphs
A decomposition of a non-empty simple graph is a pair , such that
is a set of non-empty induced subgraphs of , and every edge of
belongs to exactly one subgraph in . The chromatic index of a
decomposition is the smallest number for which there exists a
-coloring of the elements of in such a way that: for every element of
all of its edges have the same color, and if two members of share at
least one vertex, then they have different colors. A long standing conjecture
of Erd\H{o}s-Faber-Lov\'asz states that every decomposition of the
complete graph satisfies . In this paper we work
with geometric graphs, and inspired by this formulation of the conjecture, we
introduce the concept of chromatic index of a decomposition of the complete
geometric graph. We present bounds for the chromatic index of several types of
decompositions when the vertices of the graph are in general position. We also
consider the particular case in which the vertices are in convex position and
present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure
Dehn filling of cusped hyperbolic 3-manifolds with geodesic boundary
We define for each g>=2 and k>=0 a set M_{g,k} of orientable hyperbolic
3-manifolds with toric cusps and a connected totally geodesic boundary of
genus g. Manifolds in M_{g,k} have Matveev complexity g+k and Heegaard genus
g+1, and their homology, volume, and Turaev-Viro invariants depend only on g
and k. In addition, they do not contain closed essential surfaces. The
cardinality of M_{g,k} for a fixed k has growth type g^g. We completely
describe the non-hyperbolic Dehn fillings of each M in M_{g,k}, showing that,
on any cusp of any hyperbolic manifold obtained by partially filling M, there
are precisely 6 non-hyperbolic Dehn fillings: three contain essential discs,
and the other three contain essential annuli. This gives an infinite class of
large hyperbolic manifolds (in the sense of Wu) with boundary-reducible and
annular Dehn fillings having distance 2, and allows us to prove that the
corresponding upper bound found by Wu is sharp. If M has one cusp only, the
three boundary-reducible fillings are handlebodies.Comment: 28 pages, 16 figure
Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition
A geometric graph is angle-monotone if every pair of vertices has a path
between them that---after some rotation---is - and -monotone.
Angle-monotone graphs are -spanners and they are increasing-chord
graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in
2014 and proved that Gabriel triangulations are angle-monotone graphs. We give
a polynomial time algorithm to recognize angle-monotone geometric graphs. We
prove that every point set has a plane geometric graph that is generalized
angle-monotone---specifically, we prove that the half--graph is
generalized angle-monotone. We give a local routing algorithm for Gabriel
triangulations that finds a path from any vertex to any vertex whose
length is within times the Euclidean distance from to .
Finally, we prove some lower bounds and limits on local routing algorithms on
Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that
can be embedded isometrically into a hypercube, if and only if the point set
has no empty convex pentagon. Point sets of this type include convex subsets of
lattices, points on two lines, and several other infinite families. As a
consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio
A graph theoretical Poincare-Hopf Theorem
We introduce the index i(v) = 1 - X(S(v)) for critical points of a locally
injective function f on the vertex set V of a simple graph G=(V,E). Here S(v) =
{w in E | (v,w) in E, f(w)-f(v)<0} is the subgraph of the unit sphere at v in
G. It is the exit set of the gradient vector field. We prove that the sum of
i(v) over V is always is equal to the Euler characteristic X(G) of the graph G.
This is a discrete Poincare-Hopf theorem in a discrete Morse setting. It allows
to compute X(G) for large graphs for which other methods become impractical.Comment: 9 figure
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