scholarly journals Asymptotic Enumeration of Labelled Graphs by Genus

10.37236/500 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Edward A. Bender ◽  
Zhicheng Gao
Keyword(s):  
Orientable Surface ◽  
Asymptotic Formulas ◽  
Asymptotic Enumeration ◽  
Connected Graphs ◽  
Face Width ◽  
Surface Maps ◽  
Connected Surface ◽  
Labelled Graphs ◽  
Large Face

We obtain asymptotic formulas for the number of rooted 2-connected and 3-connected surface maps on an orientable surface of genus $g$ with respect to vertices and edges simultaneously. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive asymptotic formulas for the number of labelled $k$-connected graphs of orientable genus $g$ for $k\le3$.

scholarly journals The Crossing Number of a Projective Graph is Quadratic in the Face–Width

10.37236/770 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
I. Gitler ◽  
P. Hliněný ◽  
J. Leaños ◽  
G. Salazar
Keyword(s):  
Approximation Algorithm ◽  
Projective Plane ◽  
Crossing Number ◽  
Orientable Surface ◽  
Constant Factor ◽  
Bounded Degree ◽  
Face Width ◽  
Projective Graph ◽  
The Face ◽  
Large Face

We show that for each integer $g\geq0$ there is a constant $c_g > 0$ such that every graph that embeds in the projective plane with sufficiently large face–width $r$ has crossing number at least $c_g r^2$ in the orientable surface $\Sigma_g$ of genus $g$. As a corollary, we give a polynomial time constant factor approximation algorithm for the crossing number of projective graphs with bounded degree.

scholarly journals Asymptotic properties of rooted 3-connected maps on surfaces

1996 ◽  
Vol 60 (1) ◽  
pp. 31-41 ◽  
Author(s):  
Edward A. Bender ◽  
Zhicheng Gao ◽  
L. Bruce Richmond ◽  
Nicholas C. Wormald
Keyword(s):  
Asymptotic Properties ◽  
Face Width ◽  
Almost All ◽  
Large Face

AbstractIn this paper we obtain asymptotics for the number of rooted 3-connected maps on an arbitrary surface and use them to prove that almost all rooted 3-connected maps on any fixed surface have large edge-width and large face-width. It then follows from the result of Roberston and Vitray [10] that almost all rooted 3-connected maps on any fixed surface are minimum genus embeddings and their underlying graphs are uniquely embeddable on the surface.

Uniqueness and minimality of large face-width embeddings of graphs

COMBINATORICA ◽  
1995 ◽  
Vol 15 (4) ◽  
pp. 541-556 ◽  
Author(s):  
Bojan Mohar
Keyword(s):  
Face Width ◽  
Large Face

scholarly journals Asymptotic enumeration of sparse 2-connected graphs

2012 ◽  
Vol 43 (3) ◽  
pp. 354-376 ◽  
Author(s):  
Graeme Kemkes ◽  
Cristiane M. Sato ◽  
Nicholas Wormald
Keyword(s):  
Asymptotic Enumeration ◽  
Connected Graphs

XIX.—Asymptotic enumeration of connected graphs

1970 ◽  
Vol 68 (4) ◽  
pp. 298-308
Author(s):  
E. M. Wright
Keyword(s):  
Asymptotic Expansion ◽  
Generating Functions ◽  
Binomial Coefficients ◽  
Connected Components ◽  
Asymptotic Enumeration ◽  
Connected Graphs

SynopsisThe number of different connected graphs (with some property P) on n labelled nodes with q edges is fnq. Again Fnq is the number of graphs on n labelled nodes with q edges, each of whose connected components has property P. We consider 8 types of graph for which . We use a known relation between the generating functions of fnq and Fnq to find an asymptotic expansion of fnq in terms of binomial coefficients, valid if (q – ½n log n)/n→∞ as n→∞. This condition is also necessary for the existence of an asymptotic expansion of this kind.

scholarly journals The Map Asymptotics Constant $t_g$

10.37236/775 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Edward A. Bender ◽  
Zhicheng Gao ◽  
L. Bruce Richmond
Keyword(s):  
Orientable Surface ◽  
Asymptotic Formulas

The constant $t_g$ appears in the asymptotic formulas for a variety of rooted maps on the orientable surface of genus $g$. Heretofore, studying this constant has been difficult. A new recursion derived by Goulden and Jackson for rooted cubic maps provides a much simpler recursion for $t_g$ that leads to estimates for its asymptotics.

scholarly journals Counting unicellular maps on non-orientable surfaces

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Olivier Bernardi ◽  
Guillaume Chapuy
Keyword(s):  
Connected Graph ◽  
Topological Type ◽  
Orientable Surface ◽  
Recurrence Equation ◽  
Asymptotic Formulas ◽  
Topological Disk ◽  
International Audience ◽  
Fixed Topology ◽  
Orientable Surfaces

International audience A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we give a bijective operation that relates unicellular maps on a non-orientable surface to unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable precubic (all vertices of degree 1 or 3) unicellular maps of fixed topology. We also determine asymptotic formulas for the number of all unicellular maps of fixed topology, when the number of edges goes to infinity. Our strategy is inspired by recent results obtained for the orientable case [Chapuy, PTRF 2010], but significant novelties are introduced: in particular we construct an involution which, in some sense, ``averages'' the effects of non-orientability. \par Une carte unicellulaire est le plongement d'un graphe connexe dans une surface, tel que le complémentaire du graphe est un disque topologique. On décrit une opération bijective qui relie les cartes unicellulaires sur une surface non-orientable aux cartes unicellulaires de type topologique inférieur, avec des sommets marqués. On en déduit une récurrence qui conduit à de (nouvelles) formules closes d'énumération pour les cartes unicellulaires précubiques (sommets de degré 1 ou 3) de topologie fixée. On obtient aussi des formules asymptotiques pour le nombre total de cartes unicellulaires de topologie fixée, quand le nombre d'arêtes tend vers l'infini. Notre stratégie est motivée par de récents résultats dans le cas orientable [Chapuy, PTRF, 2010], mais d'importantes nouveautés sont introduites: en particulier, on construit une involution qui, en un certain sens, "moyenne'' les effets de la non-orientabilité.

scholarly journals Asymptotic Enumeration of Constellations and Related Families of Maps on Orientable Surfaces

2009 ◽  
Vol 18 (4) ◽  
pp. 477-516 ◽  
Author(s):  
GUILLAUME CHAPUY
Keyword(s):  
Graph Theory ◽  
Asymptotic Formulas ◽  
Asymptotic Enumeration ◽  
Algebraic Graph Theory ◽  
Lattice Walks ◽  
Generating Series ◽  
Finite Set ◽  
Special Case ◽  
Orientable Surfaces

We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces:m-hypermaps andm-constellations. Form= 2 they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees.Our proofs combine a bijective approach, generating series techniques related to lattice walks, and elementary algebraic graph theory.A special case of our results implies former conjectures of Z. Gao.

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