Guillaume Chapuy

Asymptotic enumeration of constellations and related families of maps on orientable surfaces

We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces: m-hypermaps and m-constellations. For m = 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.

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.

Packing Triangles in Weighted Graphs

Tuza conjectured that for every graph

Laplacian matrices and spanning trees of tree graphs

G

A bijection for covered maps on orientable surfaces

the maximum size

A Bijection for Rooted Maps on Orientable Surfaces

\nu

A new combinatorial identity for unicellular maps, via a direct bijective approach

of a set of edge-disjoint triangles and minimum size

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

\tau

A Complete Grammar for Decomposing a Family of Graphs into 3-Connected Components

of a set of edges meeting all triangles satisfy

Asymptotic enumeration and limit laws for graphs of fixed genus

\tau \leq 2\nu