A bijection for essentially 3-connected toroidal maps

by   Nicolas Bonichon, et al.

We present a bijection for toroidal maps that are essentially 3-connected (3-connected in the periodic planar representation). Our construction actually proceeds on certain closely related bipartite toroidal maps with all faces of degree 4 except for a hexagonal root-face. We show that these maps are in bijection with certain well-characterized bipartite unicellular maps. Our bijection, closely related to the recent one by Bonichon and Lévêque for essentially 4-connected toroidal triangulations, can be seen as the toroidal counterpart of the one developed in the planar case by Fusy, Poulalhon and Schaeffer, and it extends the one recently proposed by Fusy and Lévêque for essentially simple toroidal triangulations. Moreover, we show that rooted essentially 3-connected toroidal maps can be decomposed into two pieces, a toroidal part that is treated by our bijection, and a planar part that is treated by the above-mentioned planar case bijection. This yields a combinatorial derivation for the bivariate generating function of rooted essentially 3-connected toroidal maps, counted by vertices and faces.


page 1

page 2

page 3

page 4


Longer Cycles in Essentially 4-Connected Planar Graphs

A planar 3-connected graph G is called essentially 4-connected if, for e...

Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth

We present unified bijections for maps on the torus with control on the ...

Bijections between planar maps and planar linear normal λ-terms with connectivity condition

The enumeration of linear λ-terms has attracted quite some attention rec...

A Schnyder-type drawing algorithm for 5-connected triangulations

We define some Schnyder-type combinatorial structures on a class of plan...

4-connected planar graphs are in B_3-EPG

We show that every 4-connected planar graph has a B_3-EPG representation...

A bijective proof of the enumeration of maps in higher genus

Bender and Canfield proved in 1991 that the generating series of maps in...

A Study of Weisfeiler-Leman Colorings on Planar Graphs

The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure that co...

Please sign up or login with your details

Forgot password? Click here to reset