24,102 research outputs found
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Edge-disjoint double rays in infinite graphs: a Halin type result
We show that any graph that contains k edge-disjoint double rays for any k>0
contains also infinitely many edge-disjoint double rays. This was conjectured
by Andreae in 1981.Comment: 15 pages, 2 figure
On planar Cayley graphs and Kleinian groups
Let be a finitely generated group acting faithfully and properly
discontinuously by homeomorphisms on a planar surface . We prove that admits such an action that is in addition
co-compact, provided we can replace by another surface .
We also prove that if a group has a finitely generated Cayley
(multi-)graph covariantly embeddable in , then can be
chosen so as to have no infinite path on the boundary of a face.
The proofs of these facts are intertwined, and the classes of groups they
define coincide. In the orientation-preserving case they are exactly the
(isomorphism types of) finitely generated Kleinian function groups. We
construct a finitely generated planar Cayley graph whose group is not in this
class.
In passing, we observe that the Freudenthal compactification of every planar
surface is homeomorphic to the sphere
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