839 research outputs found
Hamiltonian Cycles in Polyhedral Maps
We present a necessary and sufficient condition for existence of a
contractible, non-separating and noncontractible separating Hamiltonian cycle
in the edge graph of polyhedral maps on surfaces. In particular, we show the
existence of contractible Hamiltonian cycle in equivelar triangulated maps. We
also present an algorithm to construct such cycles whenever it exists.Comment: 14 page
Hamiltonicity in multitriangular graphs
The family of 5-valent polyhedral graphs whose faces are all triangles or 3s-gons, s ≥ 9, is shown to contain non-hamiltonian graphs and to have a shortness exponent smaller than one
On cubic polyhedral graphs with prescribed adjacency properties of their faces
AbstractWe consider classes of cubic polyhedral graphs whose non-q-gonal faces are adjacent to q-gonal faces only. Structural properties of some classes of such graphs are described. For q = 5 we show that all the graphs in this class are cyclically 4-edge-connected. Some cyclically 4edge-connected and cyclically 5-edge-connected non-Hamiltonian members from this class are presented
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
We construct an exactly solvable Hamiltonian acting on a 3-dimensional
lattice of spin- systems that exhibits topological quantum order.
The ground state is a string-net and a membrane-net condensate. Excitations
appear in the form of quasiparticles and fluxes, as the boundaries of strings
and membranes, respectively. The degeneracy of the ground state depends upon
the homology of the 3-manifold. We generalize the system to , were
different topological phases may occur. The whole construction is based on
certain special complexes that we call colexes.Comment: Revtex4 file, color figures, minor correction
Computing symmetry groups of polyhedra
Knowing the symmetries of a polyhedron can be very useful for the analysis of
its structure as well as for practical polyhedral computations. In this note,
we study symmetry groups preserving the linear, projective and combinatorial
structure of a polyhedron. In each case we give algorithmic methods to compute
the corresponding group and discuss some practical experiences. For practical
purposes the linear symmetry group is the most important, as its computation
can be directly translated into a graph automorphism problem. We indicate how
to compute integral subgroups of the linear symmetry group that are used for
instance in integer linear programming.Comment: 20 pages, 1 figure; containing a corrected and improved revisio
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