20,566 research outputs found

    A note on the existence of {k, k}-equivelar polyhedral maps

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    A polyhedral map is called {p,q}\{p, q\}-equivelar if each face has pp edges and each vertex belongs to qq faces. In 1983, it was shown that there exist infinitely many geometrically realizable {p,q}\{p, q\}-equivelar polyhedral maps if q>p=4q > p = 4, p>q=4p > q = 4 or qβˆ’3>p=3q - 3 > p = 3. It was shown in 2001 that there exist infinitely many {4,4}\{4, 4\}- and {3,6}\{3, 6\}-equivelar polyhedral maps. In 1990, it was shown that {5,5}\{5, 5\}- and {6,6}\{6, 6\}-equivelar polyhedral maps exist. In this note, examples are constructed, to show that infinitely many self dual {k,k}\{k, k\}-equivelar polyhedral maps exist for each kβ‰₯5k \geq 5. Also vertex-minimal non-singular {p,p}\{p, p\}-pattern are constructed for all odd primes pp.Comment: 7 pages. To appear in `Contributions to Algebra and Geometry

    Hamiltonian Cycles in Polyhedral Maps

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    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

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    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

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    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

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    If KK is a simplicial complex on mm vertices the flagification of KK is the minimal flag complex KfK^f on the same vertex set that contains KK. Letting LL be the set of vertices, there is a sequence of simplicial inclusions Lβ†’Kβ†’KfL\to K\to K^f. This induces a sequence of maps of polyhedral products (Xβ€Ύ,Aβ€Ύ)L⟢g(Xβ€Ύ,Aβ€Ύ)K⟢f(Xβ€Ύ,Aβ€Ύ)Kf(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}. We show that Ξ©f\Omega f and Ξ©f∘Ωg\Omega f\circ\Omega g have right homotopy inverses and draw consequences. For a flag complex KK the polyhedral product of the form (CYβ€Ύ,Yβ€Ύ)K(\underline{CY},\underline Y)^K is a co-HH-space if and only if the 11-skeleton of KK is a chordal graph, and we deduce that the maps ff and f∘gf\circ g have right homotopy inverses in this case.Comment: 25 page
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