4,628 research outputs found

    Asymptotic enumeration and limit laws for graphs of fixed genus

    Full text link
    It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like c(g)n5(g−1)/2−1γnn!c^{(g)}n^{5(g-1)/2-1}\gamma^n n! where c(g)>0c^{(g)}>0, and γ≈27.23\gamma \approx 27.23 is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-connected component of linear size with high probability

    Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus

    Get PDF
    We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) 33-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the 33-connected case) to this setting. Precisely, for any cylindric essentially internally 33-connected map GG with nn vertices, we can obtain in linear time a periodic (in xx) straight-line drawing of GG that is crossing-free and internally (weakly) convex, on a regular grid Z/wZ×[0..h]\mathbb{Z}/w\mathbb{Z}\times[0..h], with w≤2nw\leq 2n and h≤n(2d+1)h\leq n(2d+1), where dd is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially 33-connected map GG on the torus (i.e., 33-connected in the periodic representation) with nn vertices, we can compute in linear time a periodic straight-line drawing of GG that is crossing-free and (weakly) convex, on a periodic regular grid Z/wZ×Z/hZ\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}, with w≤2nw\leq 2n and h≤1+2n(c+1)h\leq 1+2n(c+1), where cc is the face-width of GG. Since c≤2nc\leq\sqrt{2n}, the grid area is O(n5/2)O(n^{5/2}).Comment: 37 page

    Hamiltonian Cycles in Polyhedral Maps

    Full text link
    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
    • …
    corecore