Enumeration of maps regardless of genus: Geometric approach
نویسندگان
چکیده
We use the conceptual idea of “maps on orbifolds” and the theory of the nonEuclidian crystallographic groups (NEC groups) to enumerate rooted and unrooted maps (both sensed and unsensed) on surfaces regardless of genus. As a consequence we deduce a formula for the number of chiral pairs of maps. The enumeration principle used in this paper is due to A. D. Mednykh (2006), it counts the number of conjugacy classes of subgroups in NEC groups which are in one-two-one correspondence with unrooted (sensed or unsensed) maps.
منابع مشابه
Asymptotic Enumeration of Reversible Maps Regardless of Genus∗
We derive asymptotic expansions for the numbers U(n) of isomorphism classes of sensed maps on orientable surfaces with given number of edges n, where we do not specify the genus and for the numbers A(n) of reflexible maps with n edges. As expected the ratio A(n)/U(n) → 0 for n → ∞. This shows that almost all maps are chiral. Moreover, we show logA(n) ∼ 12 logU(n) ∼ (n/2) log n. Due to a corresp...
متن کاملRooted maps on orientable surfaces, Riccati's equation and continued fractions
We present a new approach in the study of rooted maps without regard to genus. We prove the existence of a new type of equation for the generating series of these maps enumerated with respect to edges and vertices. This is Riccati’s equation. It seems to be the first time that such a differential equation appears in the enumeration of rooted maps. Solving this equation leads to different closed...
متن کاملCharacter Theory and Rooted Maps in an Orientable Surface of given Genus: Face-colored Maps
The character theoretic approach [5] to the enumeration of rooted maps in an orientable surface of arbitrary genus is extended to 2-face-colorable rooted maps. In particular, we show that there exists, for each genus, a correspondence between the set of 2-colored triangulations and a set of 2-colored rooted maps of all lower genera with a distinguished subset of vertices.
متن کاملA bijective proof of the enumeration of maps in higher genus
Bender and Canfield proved in 1991 that the generating series of maps in higher genus is a rational function of the generating series of planar maps. In this paper, we give the first bijective proof of this result. Our approach starts with the introduction of a canonical orientation that enables us to construct a bijection between 4-valent bicolorable maps and a family of unicellular blossoming...
متن کاملA new approach for enumerating maps on orientable surfaces
Classifying embeddings of a given graph G on orientable surfaces under the action of its automorphisms, a relation between the genus distribution of rooted maps and embeddings of graph G on orientable surfaces is established. Applying this relation enables us to enumerate rooted maps by automorphism groups or by enumerating labelled graphs with vertex partition and find new formulas for the num...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics
دوره 310 شماره
صفحات -
تاریخ انتشار 2010