The Semi-Arc Automorphism Group of a Graph with Application to Map Enumeration

A map is a connected topological graph cellularly embedded in a surface. For a given graph Γ, its genus distribution of rooted maps and embeddings on orientable and non-orientable surfaces are separately investigated by many researchers. By introducing the concept of a semi-arc automorphism group of a graph and classifying all its embeddings under the action of its semi-arc automorphism group, we find the relations between its genus distribution of rooted maps and genus distribution of embeddings on orientable and non-orientable surfaces, and give some new formulas for the number of rooted maps on a given orientable surface with underlying graph a bouquet of cycles B_n, a closed-end ladder L_n or a Ringel ladder R_n. A general scheme for enumerating unrooted maps on surfaces(orientable or non-orientable) with a given underlying graph is established. Using this scheme, we obtained the closed formulas for the numbers of non-isomorphic maps on orientable or non-orientable surfaces with an underlying bouquet B_n in this paper.     

An analogue of the Harer-Zagier formula for unicellular maps on general surfaces

A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is simply connected. In a famous article, Harer and Zagier established a formula for the generating function of unicellular maps counted according to the number of vertices and edges. The keystone of their approach is a counting formula for unicellular maps on orientable surfaces with n edges, and with vertices colored using every color in [q] (adjacent vertices are authorized to have the same color). We give an analogue of this formula for general (locally orientable) surfaces.Our approach is bijective and is inspired by Lass?s proof of the Harer-Zagier formula. We first revisit Lass?s proof and twist it into a bijection between unicellular maps on orientable surfaces with vertices colored using every color in [q], and maps with vertex set [q] on orientable surfaces with a marked spanning tree. The bijection immediately implies Harer-Zagier?s formula and a formula by Jackson concerning bipartite unicellular maps. It also shed a new light on constructions by Goulden and Nica, Schaeffer and Vassilieva, and Morales and Vassilieva. We then extend the bijection to general surfaces and obtain a correspondence between unicellular maps on general surfaces with vertices colored using every color in [q], and maps on orientable surfaces with vertex set [q]with a marked planar submap. This correspondence gives an analogue of the Harer-Zagier formula for general surfaces. We also show that this formula implies a recursion formula due to Ledoux for the numbers of unicellular maps with given numbers of vertices and edges.     

Group action for enumerating maps on surfaces

A map is a connected topological graph Γ cellularly embedded in a surface. For any connected graph Γ, by introducing the conception of semi-arc automorphism groupAut1/2 Γ and classifying all embedding of Γ under the action of this group, the numbersr ^o (Γ) andr ^N (Γ) of rooted maps on orientable and non-orientable surfaces with underlying graph Γ are found. Many closed formulas without sum Σ for the number of rooted maps on surfaces (orientable or non-orientable) with given underlying graphs, such as, complete graphK _n , complete bipartite graphK _m,n, bouquetsB _n , dipoleDp _n and generalized dipoleDp _n ^k,l are refound in this paper.     

Surface embeddings of Steiner triple systems

A Steiner triple system of order n (STS(n)) is said to be embeddable in an orientable surface if there is an orientable embedding of the complete graph K_n whose faces can be properly 2-colored (say, black and white) in such a way that all black faces are triangles and these are precisely the blocks of the STS(n). If, in addition, all white faces are triangular, then the collection of all white triangles forms another STS(n); the pair of such STS(n)s is then said to have an (orientable) bi-embedding. We study several questions related to embeddings and bi-embeddings of STSs. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 325–336, 1998     

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.     

Annular embeddings of permutations for arbitrary genus

In the symmetric group on a set of size 2n, let P2n denote the conjugacy class of involutions with no fixed points (equivalently, we refer to these as “pairings”, since each disjoint cycle has length 2). Harer and Zagier explicitly determined the distribution of the number of disjoint cycles in the product of a fixed cycle of length 2n and the elements of P2n. Their famous result has been reproved many times, primarily because it can be interpreted as the genus distribution for 2-cell embeddings in an orientable surface, of a graph with a single vertex attached to n loops. In this paper we give a new formula for the cycle distribution when a fixed permutation with two cycles (say the lengths are p,q, where p+q=2n) is multiplied by the elements of P2n. It can be interpreted as the genus distribution for 2-cell embeddings in an orientable surface, of a graph with two vertices, of degrees p and q. In terms of these graphs, the formula involves a parameter that allows us to specify, separately, the number of edges between the two vertices and the number of loops at each of the vertices. The proof is combinatorial, and uses a new algorithm that we introduce to create all rooted forests containing a given rooted forest.     

Classification of regular embeddings of hypercubes of odd dimension

By a regular embedding of a graph into a closed surface we mean a 2-cell embedding with the automorphism group acting regularly on flags. Recently, Kwon and Nedela [Non-existence of nonorientable regular embeddings of n-dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the n-dimensional cubes Q_n into nonorientable surfaces exist for any positive integer n>2. In 1997, Nedela and Škoviera [Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997) 807-823] presented a construction giving for each solution of the congruence a regular embedding M_e of the hypercube Q_n into an orientable surface. It was conjectured that all regular embeddings of Q_n into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes Q_n into orientable surfaces for n odd, proving affirmatively the conjecture of Nedela and Škoviera for every odd n.     

A bijection for covered maps, or a shortcut between Harer-Zagier?s and Jackson?s formulas

We consider maps on orientable surfaces. A map is called unicellular if it has a single face. A covered map is a map (of genus g) with a marked unicellular spanning submap (which can have any genus in {0,1,…,g}). Our main result is a bijection between covered maps with n edges and genus g and pairs made of a plane tree with n edges and a unicellular bipartite map of genus g with n+1 edges. In the planar case, covered maps are maps with a marked spanning tree and our bijection specializes into a construction obtained by the first author in Bernardi (2007) [4].Covered maps can also be seen as shuffles of two unicellular maps (one representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of unicellular maps, and pairs made of a plane tree and a unicellular bipartite map. In terms of counting, this establishes the equivalence between a formula due to Harer and Zagier for general unicellular maps, and a formula due to Jackson for bipartite unicellular maps.We also show that the bijection of Bouttier, Di Francesco and Guitter (2004) [8] (which generalizes a previous bijection by Schaeffer, 1998 [33]) between bipartite maps and so-called well-labeled mobiles can be obtained as a special case of our bijection.     

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.     

The asymptotic number of rooted maps on a surface

Let S be a surface. We asymptotically enumerate two classes of n-edged maps on S as n → ∞: rooted and rooted smooth. These are based on a system of equations which give, in principle, the exact generating functions.     

The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees

A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of given genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer’s bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ISE random measure, and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ISE.     

Counting colored planar maps: Algebraicity results

We address the enumeration of properly q-colored planar maps, or more precisely, the enumeration of rooted planar maps M weighted by their chromatic polynomial χM(q) and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when q≠0,4 is of the form 2+2cos(jπ/m), for integers j and m. This includes the two integer values q=2 and q=3. We extend this to planar maps weighted by their Potts polynomial PM(q,ν), which counts all q-colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper q-colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two “catalytic” variables. To our knowledge, this is the first time such equations are being solved since Tutte?s remarkable solution of properly q-colored triangulations.     

Regular orientable imbeddings of complete graphs

This paper classifies the regular imbeddings of the complete graphs K_n in orientable surfaces. Biggs showed that these exist if and only if n is a prime power p^e, his examples being Cayley maps based on the finite field F = GF(n). We show that these are the only examples, and that there are $\text{φ(n ? 1)}\text{e}$ isomorphism classes of such maps (where φ is Euler's function), each corresponding to a conjugacy class of primitive elements of F, or equivalently to an irreducible factor of the cyclotomic polynomial Φ_{n ? 1}(z) over GF(p). We show that these maps are all equivalent under Wilson's map-operations H_i, and we determined for which n they are reflexible or self-dual.     

New regular embeddings of n‐cubes Qn

We investigate highly symmetrical embeddings of the n‐dimensional cube Q_n into orientable compact surfaces which we call regular embeddings of Q_n. We derive some general results and construct some new families of regular embeddings of Q_n. In particular, for n = 1, 2,3(mod 4), we classify the regular embeddings of Q_n which can be expressed as balanced Cayley maps. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 297–312, 2004     

Special manifolds and shape fibrator properties

Our main interest in this paper is further investigation of the concept of (PL) fibrators (introduced by Daverman [R.J. Daverman, PL maps with manifold fibers, J. London Math. Soc. (2) 45 (1992) 180-192]), in a slightly different PL setting. Namely, we are interested in manifolds that can detect approximate fibrations in the new setting. The main results state that every orientable, special (a new class of manifolds that we introduce) PL n-manifold with non-trivial first homology group is a fibrator in the new category, if it is a codimension-2 fibrator (Theorem 8.2) or has a non-cyclic fundamental group (Theorem 8.4). We show that all closed, orientable surface S with χ(S)<0 are fibrators in the new category.     

Higher-dimensional Bruckner-Garg type theorem

In this paper, we investigate several properties of maps from a compactum X to an n-dimensional (combinatorial) manifold Mn. We introduce the notions of stable point and locally extreme point of map, and we prove a higher-dimensional Bruckner-Garg type theorem for the fiber structure of a generic map in the space C(X,Mn) of maps from a compactum X with dimX?n to an n-dimensional manifold Mn (n?1). As applications, we also study the spaces of Bing maps, Lelek maps, k-dimensional maps and Krasinkiewicz maps in C(X,Mn).     

Non-orientable Regular Maps of a Given Type over Linear Fractional Groups

It is known that for any given k and m such that 1/k + 1/m < 1/2 there exist infinitely many regular maps M of valence k and face length m on orientable surfaces such that the automorphism group of M is isomorphic to a linear fractional group over a finite field. We examine the pairs (k, m) for which this result can be extended to regular maps on non-orientable surfaces.     

You can hear the local orientability of an orbifold

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold which is locally modeled on the quotient of a connected, open manifold under a finite group of isometries. If all of the isometries used to define the local structures of an entire orbifold are orientation preserving, we call the orbifold locally orientable. We use heat invariants to show that a Riemannian orbifold which is locally orientable cannot be Laplace isospectral to a Riemannian orbifold which is not locally orientable. As a corollary we observe that a Riemannian orbifold that is not locally orientable cannot be Laplace isospectral to a Riemannian manifold.     

Oriented matroids and complete-graph embeddings on surfaces

We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable and when one color class of cells consists of triangles only. Precisely for triangular pseudoline arrangements, one element of the image pair of f is a triangular complete-graph embedding on a surface. We obtain all triangular complete-graph embeddings on surfaces this way, when we extend the definition of triangular complete pseudoline arrangements in a natural way to that of triangular curve arrangements on surfaces in which each pair of curves has a point in common where they cross. Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces. Furthermore, we establish the relationship between 2-colorable curve arrangements and Petrie dual maps. A data structure, called intersection pattern is provided for the study of curve arrangements on surfaces. Finally we show that an orientable surface of genus g admits a complete curve arrangement with at most 2g+1 curves in contrast to the non-orientable surface where the number of curves is not bounded.     

Rigidity for orientable functors

In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K-theory of algebraically closed fields. Besides K-theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes.