ENUMERATING ROOTED LOOPLESS PLANAR MAPS

This paper provides the following results.1.The equivalence between the method described by W.T.Tutte for determining parametricexpressions of certain enumerating functions and the one which the author used in [2] for findingthe parametric expression of the generating function of rooted general planar maps dependent on theedge number,is shown.2.The number of rooted boundary loop maps,i.e.,maps for each of which all the edges on theboundary of the outer face are loops,with the edge number given is found.3.The number of rooted nearly loopless planar maps,i.e.,loopless maps and maps having exactlyone loop which is just the rooted edge and does not form the boundary of the outer face,with givenedge number is also found.4.The recursive formula satisfied by the number of rooted loopless planar maps dependent onthe edge number is derived.5.In addition,the number of loop rooted maps,i.e.,maps in each of which there is only one loopwhich is just the rooted edge,dependent on the edge number is obtained at the sam     

Counting rooted unicursal planar maps

This paper investigates the number of rooted unicursal planar maps and presents some formulae for such maps with four parameters： the numbers of nonrooted vertices and inner faces and the valencies of two odd vertices.     

Enumeration of rooted planar halin maps

A Halin map is a kind of planar maps oriented by a tree. In this paper the rooted halin maps with the vertex partition as parameters are enumerated such that a famous result on rooted trees due to Harary. Prins, and Tutte is deduced as. a special ease. Further, by using Lagrangian inversion to obtain a number of summation free formulae dixectly, the various kinds of rooted Halin maps with up to three parameters have been counted.     

Counting rooted 4-regular unicursal planar maps

A map is 4-regular unicursal if all its vertices are 4-valent except two odd-valent vertices. This paper investigates the number of rooted 4-regular unicursal planar maps and presents some formulae for such maps with four parameters: the number of edges, the number of inner faces and the valencies of the two odd vertices.     

The number of rooted Eulerian planar maps

In this paper we provide a solution of the functional equation unsolved in the paper, by the second author, "On functional equations arising from map enumerations" that appeared in Discrete Math, 123: 93-109 (1993). It is also the number of combinatorial distinct rooted general eulerian planar maps with the valency of root-vertex, the number of non-root vertices and non-root faces of the maps as three parameters. In particular, a result in the paper, by the same author, "On the number of eulerian planar map...     

Chromatic sum equations for rooted cubic planar maps

This paper provides a functional equation satisfied by rooted nearly cubic planar maps. By a nearly cubic map is meant such a map that all the vertices have valency 3 with the exception of at most the root-vertex. And, as a consequence, the corresponding functional equation for rooted cubic planar maps is found.     

A census of boundary cubic rooted planar maps

In this paper, boundary cubic rooted planar maps are investigated and exact enumerative formulae are given. First, an enumerative formula for boundary cubic inner-forest maps with the size (number of edges) as a parameter is derived. For the special case of boundary cubic inner-tree maps, a simple formula with two parameters is presented. Further, according to the duality, a corresponding result for outer-planar maps is obtained. Finally, some results for boundary cubic planar maps and general planar maps are obtained. Furthermore, two known Tutte's formulae are easily deduced in the paper.     

Enumeration of multi-colored rooted maps

We present a study of n-colored rooted maps in orientable and locally orientable surfaces. As far as we know, no work on these maps has yet been published. We give a system of n functional equations satisfied by n-colored orientable rooted maps regardless of genus and with respect to edges and vertices. We exhibit the solution of this system as a vector where each component has a continued fraction form and we deduce a new equation generalizing the Dyck equation for rooted planar trees. Similar results are shown for n-colored rooted maps in locally orientable surfaces.     

Enumeration of rooted nonseparable outerplanar maps

In this paper, the number of combinatorially distinct rooted nonseparable outerplanar maps withm edges and the valency of the root-face being n is found to be(m-1)! (m-2) !:(n-1)!(n-2)! (m-n)!(m-n 1)!and, the number of rooted nonseparable outerplanar maps with m edges is also determined to be(2m-2)!:(m-1)!m!,which is just the number of distinct rooted plane trees with m-1 edges.     

Counting rooted maps by genus III: Nonseparable maps

A map on an orientable surface is called separable if its underlying graph can be disconnected by splitting a vertex into two pieces, each containing a positive number of edge-ends consecutively ordered with respect to counter-clockwise rotation around the original vertex. This definition is shown to be equivalent for planar maps to Tutte's definition of a separable planar map. We develop a procedure for determining the generating functions N_g,b(x) = Σ_p=0^∞n_g,b,px^p, where n_g,b,p is the number of rooted nonseparable maps with b + p edges and p + 1 vertices on an orientable surface of genus g. Similar results are found for tree-rooted maps.     

Almost all rooted maps have large representativity

Let M be a map on a surface S. The edge-width of M is the length of a shortest noncontractible cycle of M. The face-width (or, representativity) of M is the smallest number of intersections a noncontractible curve in S has with M. (The edge-width and face-width of a planar map may be defined to be infinity.) A map is a large-edge-width embedding (LEW-embedding) if its maximum face valency is less than its edge-width. For several families of rooted maps on a given surface, we prove that there are positive constants C₁ and C₂, depending on the family and the surface, such that

• 1 almost all maps with n edges have face-width and edge-width greater than c₁ log n, and
• 2 the fraction of such maps that are LEW-embeddings and the fraction that are not LEW-embeddings both exceed n^{? >C2}.

     

The number of rooted circuit boundary maps

In this article, the authors discuss two kinds of new planar maps： pan-fan maps and circuit boundary maps, and provide explicit expressions about their enumerating functions with different parameters. Meanwhile, two explicit counting formulas for circuit cubic boundary maps with two parameters; the size and the valency of the root-face, are also extracted.     

Counting rooted maps by Genus. I

Using a combinatorial equivalent for maps, we take the first census of maps on orientable surfaces of arbitrary genus. We generalize to higher genus Tutte's recursion formula for counting slicings, and thus obtain an algorithm for counting rooted maps by genus, number of edges, and number of vertices. We then solve a special case of this recursion formula, to count slicngs with one face by genus. This leads to an explicit formula which counts rooted maps with one face by genus and number of edges.     

Matching planar maps

The subject of this paper are algorithms for measuring the similarity of patterns of line segments in the plane, a standard problem in, e.g., computer vision, geographic information systems, etc. More precisely, we define feasible distance measures that reflect how close a given pattern H is to some part of a larger pattern G. These distance measures are generalizations of the well-known Fréchet distance for curves. We first give an efficient algorithm for the case that H is a polygonal curve and G is a geometric graph. Then, slightly relaxing the definition of distance measure, we give an algorithm for the general case where both, H and G, are geometric graphs.     

The number of rooted essential maps on surfaces

In this paper, general functional equations of rooted essential maps on surfaces （orientable and nonorientable） are deduced and their formal solutions are presented. Further, three explicit for- mulae for counting essential maps on S2,N3 andN4 are given. In the same time, some known results can be derived.     

Counting unrooted planar maps

Two planar maps are identified if one can be transformed to the other by any homeomorphism of the sphere. The number of such maps is found by determining the numbers of maps which have been rooted in various ways.