20,566 research outputs found
A note on the existence of {k, k}-equivelar polyhedral maps
A polyhedral map is called -equivelar if each face has edges
and each vertex belongs to faces. In 1983, it was shown that there exist
infinitely many geometrically realizable -equivelar polyhedral maps
if , or . It was shown in 2001 that there
exist infinitely many - and -equivelar polyhedral maps. In
1990, it was shown that - and -equivelar polyhedral maps
exist. In this note, examples are constructed, to show that infinitely many
self dual -equivelar polyhedral maps exist for each . Also
vertex-minimal non-singular -pattern are constructed for all odd
primes .Comment: 7 pages. To appear in `Contributions to Algebra and Geometry
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
Contractible Hamiltonian Cycles in Polyhedral Maps
We present a necessary and sufficient condition for existence of a
contractible Hamiltonian Cycle in the edge graph of equivelar maps on surfaces.
We also present an algorithm to construct such cycles. This is further
generalized and shown to hold for more general maps.Comment: 9 pages, 1 figur
On generating a diminimal set of polyhedral maps on the torus
We develop a method to find a set of diminimal polyhedral maps on the torus
from which all other polyhedral maps on the torus may be generated by face
splitting and vertex splitting. We employ this method, though not to its
completion, to find 53 diminimal polyhedral maps on the Torus
The homotopy theory of polyhedral products associated with flag complexes
If is a simplicial complex on vertices the flagification of is
the minimal flag complex on the same vertex set that contains .
Letting be the set of vertices, there is a sequence of simplicial
inclusions . This induces a sequence of maps of polyhedral
products .
We show that and have right homotopy
inverses and draw consequences. For a flag complex the polyhedral product
of the form is a co--space if and only if
the -skeleton of is a chordal graph, and we deduce that the maps and
have right homotopy inverses in this case.Comment: 25 page
- β¦