A Note on the Existence of {k, k}-equivelar Polyhedral Maps
نویسنده
چکیده
A polyhedral map is called {p, q}-equivelar if each face has p edges and each vertex belongs to q faces. In [12], it was shown that there exist infinitely many geometrically realizable {p, q}-equivelar polyhedral maps if q > p = 4, p > q = 4 or q − 3 > p = 3. It was shown in [6] that there exist infinitely many {4, 4}and {3, 6}-equivelar polyhedral maps. In [1], it was shown that {5, 5}and {6, 6}-equivelar polyhedral maps exist. In this note, examples are constructed, to show that infinitely many self dual {k, k}-equivelar polyhedral maps exist for each k ≥ 5. Also vertex-minimal non-singular {p, p}-patterns are constructed for all odd primes p. MSC 2000: 52B70, 51M20, 57M20
منابع مشابه
3 0 Ju n 20 05 A note on the existence of { k , k } - equivelar polyhedral maps
A polyhedral map is called {p, q}-equivelar if each face has p edges and each vertex belongs to q faces. In [12], it was shown that there exist infinitely many geometrically realizable {p, q}-equivelar polyhedral maps if q > p = 4, p > q = 4 or q − 3 > p = 3. It was shown in [6] that there exist infinitely many {4, 4}and {3, 6}-equivelar polyhedral maps. In [1], it was shown that {5, 5}and {6, ...
متن کاملEquivelar maps on the torus
We give a classification of all equivelar polyhedral maps on the torus. In particular, we classify all triangulations and quadrangulations of the torus admitting a vertex transitive automorphism group. These are precisely the ones which are quotients of the regular tessellations {3, 6}, {6, 3} or {4, 4} by a pure translation group. An explicit formula for the number of combinatorial types of eq...
متن کاملContractible 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. AMS classification : 57Q15, 57M20, 57N05.
متن کاملThe Edge - minimal Polyhedral Maps of Euler Characteristic – 8 ∗
In [2], a {5, 5}-equivelar polyhedral map of Euler characteristic −8 was constructed. In this article we prove that {5, 5}-equivelar polyhedral map of Euler characteristic −8 is unique. As a consequence, we get that the minimum number of edges in a non-orientable polyhedral map of Euler characteristic −8 is > 40. We have also constructed {5, 5}-equivelar polyhedral map of Euler characteristic −...
متن کاملGeneralized Continuous Frames for Operators
In this note, the notion of generalized continuous K- frame in a Hilbert space is defined. Examples have been given to exhibit the existence of generalized continuous $K$-frames. A necessary and sufficient condition for the existence of a generalized continuous $K$-frame in terms of its frame operator is obtained and a characterization of a generalized continuous $K$-frame for $ mathcal{H} $ wi...
متن کامل