57 research outputs found
Irreducible Quadrangulations of the Torus
AbstractIn this paper, we find the irreducible quadrangulations of the torus. As a consequence, any two quadrangulations of the torus with the same number of vertices that are either both bipartite or both non-bipartite (except for some complete bipartite graphs) can be transformed into one another, up to homeomorphism, using a sequence of diagonal slides and diagonal rotations. We also determine the minor minimal 2-representative graphs on the torus
Asymptotic enumeration and limit laws for graphs of fixed genus
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
where , and 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
Basic nets in the projective plane
The notion of basic net (called also basic polyhedron) on plays a
central role in Conway's approach to enumeration of knots and links in .
Drobotukhina applied this approach for links in using basic nets on
. By a result of Nakamoto, all basic nets on can be obtained from a
very explicit family of minimal basic nets (the nets , ,
in Conway's notation) by two local transformations. We prove a similar result
for basic nets in .
We prove also that a graph on is uniquely determined by its pull-back
on (the proof is based on Lefschetz fix point theorem).Comment: 14 pages, 15 figure
A generalization of the quadrangulation relation to constellations and hypermaps
Constellations and hypermaps generalize combinatorial maps, i.e. embedding of
graphs in a surface, in terms of factorization of permutations. In this paper,
we extend a result of Jackson and Visentin (1990) stating an enumerative
relation between quadrangulations and bipartite quadrangulations. We show a
similar relation between hypermaps and constellations by using a result of
Littlewood on factorization of characters. A combinatorial proof of
Littlewood's result is also given. Furthermore, we show that coefficients in
our relation are all positive integers, hinting possibility of a combinatorial
interpretation. Using this enumerative relation, we recover a result on the
asymptotic behavior of hypermaps in Chapuy (2009).Comment: 19 pages, extended abstract published in the proceedings of FPSAC
201
Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
Let be a connected closed oriented surface of genus . Given a
triangulation (resp. quadrangulation) of , define the index of each of its
vertices to be the number of edges originating from this vertex minus
(resp. minus ). Call the set of integers recording the non-zero indices the
profile of the triangulation (resp. quadrangulation). If is a profile
for triangulations (resp. quadrangulations) of , for any , denote by (resp.
) the set of (equivalence classes of) triangulations
(resp. quadrangulations) with profile which contain at most
triangles (resp. squares). In this paper, we will show that if is a
profile for triangulations (resp. for quadrangulations) of such that none
of the indices in is divisible by (resp. by ), then
(resp.
), where and . The key ingredient of the proof is a
result of J. Koll\'ar on the link between the curvature of the Hogde metric on
vector subbundles of a variation of Hodge structure over algebraic varieties,
and Chern classes of their extensions. By the same method, we also obtain the
rationality (up to some power of ) of the Masur-Veech volume of arithmetic
affine submanifolds of translation surfaces that are transverse to the kernel
foliation.Comment: 24 pages, to appear in Journal de l'Ecole Polytechnique:
Math\'ematique
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