scholarly journals Counting 2-Connected 4-Regular Maps on the Projective Plane

10.37236/4038 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Shude Long ◽  
Han Ren
Keyword(s):  
Projective Plane ◽  
Regular Maps ◽  
Asymptotic Formulae ◽  
Special Cases ◽  
Planar Maps ◽  
Darboux's Method

In this paper the number of rooted (near-) 4-regular maps on the projective plane are investigated with respect to the root-valency, the number of edges, the number of inner faces, the number of nonroot-vertex-loops, the number of nonroot-vertex-blocks. As special cases, formulae for several types of rooted 4-regular maps such as 2-connected 4-regular projective planar maps, rooted 2-connected (connected) 4-regular projective planar maps without loops are also presented. Several known results on the number of 4-regular maps on the projective plane are also concluded. Finally, by use of Darboux's method, very nice asymptotic formulae for the numbers of those types of maps are given.

Enumeration of unsensed r-regular maps on the projective plane and the Klein bottle

2021 ◽  
Vol 344 (11) ◽  
pp. 112528
Author(s):  
Evgeniy Krasko ◽  
Alexander Omelchenko
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Regular Maps

scholarly journals A Generic Method for Bijections between Blossoming Trees and Planar Maps

10.37236/3386 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Marie Albenque ◽  
Dominique Poulalhon
Keyword(s):  
Recent Work ◽  
Spanning Tree ◽  
Linear Time ◽  
Computational Scheme ◽  
Closed Formula ◽  
Special Cases ◽  
Planar Maps ◽  
Root Face ◽  
By Products

This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of application so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face.The bijective construction presented here relies deeply on the theory of $\alpha$-orientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases many previously known bijections involving blossoming trees: for example Eulerian maps, $m$-Eulerian maps, non-separable maps and simple triangulations and quadrangulations of a $k$-gon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and $d$-angulations of girth $d$ of a $k$-gon.As for applications, each specialization of the construction translates into enumerative by-products, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode or sample planar maps.In a recent work, Bernardi and Fusy introduced another unified bijective scheme; we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.

An Easy Proof for Some Classical Theorems in Plane Geometry

1992 ◽  
Vol 35 (4) ◽  
pp. 560-568 ◽  
Author(s):  
C. Thas
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
The Other ◽  
Plane Geometry ◽  
The Real ◽  
Easy Proof ◽  
Special Cases ◽  
Euclidean Planes ◽  
Straight Lines ◽  
Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

scholarly journals Introducing edge-biregular maps

2021 ◽  
Author(s):  
Olivia Reade
Keyword(s):  
Automorphism Group ◽  
Euler Characteristic ◽  
Dihedral Group ◽  
Automorphism Groups ◽  
Triangle Group ◽  
Supporting Surface ◽  
Regular Maps ◽  
Special Cases ◽  
Edge Colourings

AbstractWe introduce the concept of alternate-edge-colourings for maps and study highly symmetric examples of such maps. Edge-biregular maps of type (k, l) occur as smooth normal quotients of a particular index two subgroup of $$T_{k,l}$$ T k , l , the full triangle group describing regular plane (k, l)-tessellations. The resulting colour-preserving automorphism groups can be generated by four involutions. We explore special cases when the usual four generators are not distinct involutions, with constructions relating these maps to fully regular maps. We classify edge-biregular maps when the supporting surface has non-negative Euler characteristic, and edge-biregular maps on arbitrary surfaces when the colour-preserving automorphism group is isomorphic to a dihedral group.

On Baer Involutions of Finite Projective Planes

1970 ◽  
Vol 22 (4) ◽  
pp. 878-880 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Finite Projective Planes ◽  
Special Cases ◽  
Collineation Groups ◽  
Baer Involution ◽  
Finite Desarguesian Projective Plane

1. An involution of a projective plane π is a collineation X of π such that λ2 = 1. Involutions play an important röle in the theory of finite projective planes. According to Baer [2], an involution λ of a finite projective plane of order n is either a perspectivity, or it fixes a subplane of π of order in the last case, λ is called a Baer involution.While there are many facts known about collineation groups of finite projective planes containing perspectivities (see for instance [4; 5]), the investigation of Baer involutions seems rather difficult. The few results obtained about planes admitting Baer involutions are restricted only to special cases. Our aim in the present paper is to investigate finite projective planes admitting a large number of Baer involutions. It is known (see for instance [3, p. 401]) that in a finite Desarguesian projective plane of square order, the vertices of every quadrangle are fixed by exactly one Baer involution.

Counting Coloured Graphs of High Connectivity

1981 ◽  
Vol 33 (2) ◽  
pp. 476-484 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Graph Theory ◽  
Functional Equations ◽  
Bipartite Graphs ◽  
Asymptotic Expressions ◽  
Asymptotic Formulae ◽  
Special Cases ◽  
High Connectivity ◽  
Special Case ◽  
Labelled Graphs

Find exact or asymptotic formulae for the number of labelled graphs of order n having a certain property. The property we are interested in in this note is that of being k-coloured and having connectivity at least l. Special cases of this problem have been tackled by many authors; in particular Gilbert [6], Read [9] and Robinson [11] found exact formulae, and Read and Wright [10], and Wright [12], [13] found asymptotic expressions (for many other examples see [7]). Recently Harary and Robinson [8] counted labelled bipartite blocks, that is 2-connected bipartite graphs. (For terms not defined here and general background in graph theory see [1].) Our present investigations have been prompted by [8]; in particular, as a very special case of our results, we shall prove the conjecture published in [8].The exact formulae appearing in the enumeration of labelled graphs in general, and in the enumeration of k-coloured labelled graphs in particular, tend to be very pleasing, especially because of the functional equations relating them.

scholarly journals The fourth moment of individual Dirichlet L-functions on the critical line

2020 ◽  
Author(s):  
Berke Topacogullari
Keyword(s):  
Critical Line ◽  
Power Saving ◽  
Zeta Functions ◽  
Number Fields ◽  
Fourth Moment ◽  
Quadratic Number ◽  
Second Moment ◽  
Dirichlet Characters ◽  
Asymptotic Formulae ◽  
Special Cases

Abstract We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line.

Measurement of area by counting

1965 ◽  
Vol 2 (02) ◽  
pp. 339-351 ◽  
Author(s):  
Allan M. Russell ◽  
Nora S. Josephson
Keyword(s):  
General Formula ◽  
Analytical Methods ◽  
Regular Point ◽  
Geometrical Probability ◽  
Asymptotic Formulae ◽  
Special Cases ◽  
Computer Calculations ◽  
Planar Figure ◽  
Point Lattice

Summary Through the use of a zone mapping transformation, geometrical probability is applied to the problem of measuring the area of a planar figure by counting the points covered when the figure is placed at random on a regular point lattice. An expression for the variance σ 2 is obtained as a function of radius for the case of a circle. This expression is equivalent to a more general formula for σ 2 derived earlier by Kendall and Rankin using analytical methods. Computer calculations of σ are given for two special cases and the results are compared with those predicted by alternative asymptotic formulae also due to Kendall and Rankin. Some examples of the use of zone mapping to treat asymmetric figures, not subject to analytical methods, are described.

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