3,096 research outputs found
Recognizing halved cubes in a constant time per edge
AbstractGraphs that can be isometrically embedded into the metric space l1 are called l1-graphs. Halved cubes play an important role in the characterization of l1-graphs. We present an algorithm that recognizes halved cubes in O(n log2 n) time
Primary Non-QE Graphs on Six Vertices
A connected graph is called of non-QE class if it does not admit a quadratic
embedding in a Euclidean space. A non-QE graph is called primary if it does not
contain a non-QE graph as an isometrically embedded proper subgraph. The graphs
on six vertices are completely classified into the classes of QE graphs, of
non-QE graphs, and of primary non-QE graphs.Comment: arXiv admin note: text overlap with arXiv:2206.0584
Characterizations of the Suzuki tower near polygons
In recent work, we constructed a new near octagon from certain
involutions of the finite simple group and showed a correspondence
between the Suzuki tower of finite simple groups, , and the tower of near polygons, . Here we characterize
each of these near polygons (except for the first one) as the unique near
polygon of the given order and diameter containing an isometrically embedded
copy of the previous near polygon of the tower. In particular, our
characterization of the Hall-Janko near octagon is similar to an
earlier characterization due to Cohen and Tits who proved that it is the unique
regular near octagon with parameters , but instead of regularity
we assume existence of an isometrically embedded dual split Cayley hexagon,
. We also give a complete classification of near hexagons of
order and use it to prove the uniqueness result for .Comment: 20 pages; some revisions based on referee reports; added more
references; added remarks 1.4 and 1.5; corrected typos; improved the overall
expositio
Embedding of metric graphs on hyperbolic surfaces
An embedding of a metric graph on a closed hyperbolic surface is
\emph{essential}, if each complementary region has a negative Euler
characteristic. We show, by construction, that given any metric graph, its
metric can be rescaled so that it admits an essential and isometric embedding
on a closed hyperbolic surface. The essential genus of is the
lowest genus of a surface on which such an embedding is possible. In the next
result, we establish a formula to compute . Furthermore, we show that
for every integer , admits such an embedding (possibly
after a rescaling of ) on a surface of genus .
Next, we study minimal embeddings where each complementary region has Euler
characteristic . The maximum essential genus of is
the largest genus of a surface on which the graph is minimally embedded.
Finally, we describe a method explicitly for an essential embedding of , where and are realized.Comment: Revised version, 11 pages, 3 figure
Coarse geometry of the fire retaining property and group splittings
Given a non-decreasing function we
define a single player game on (infinite) connected graphs that we call fire
retaining. If a graph admits a winning strategy for any initial
configuration (initial fire) then we say that has the -retaining
property; in this case if is a polynomial of degree , we say that
has the polynomial retaining property of degree .
We prove that having the polynomial retaining property of degree is a
quasi-isometry invariant in the class of uniformly locally finite connected
graphs. Henceforth, the retaining property defines a quasi-isometric invariant
of finitely generated groups. We prove that if a finitely generated group
splits over a quasi-isometrically embedded subgroup of polynomial growth of
degree , then has polynomial retaining property of degree . Some
connections to other work on quasi-isometry invariants of finitely generated
groups are discussed and some questions are raised.Comment: 16 pages, 1 figur
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