Geodesic
Bijective Proof of the Rationality of the Generating Series of Higher-Genus Maps
Séminaire lotharingien de combinatoire, 80B (2018) 
Mathias Lepoutre. Bijective Proof of the Rationality of the Generating Series of Higher-Genus Maps. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a66/
@article{SLC_2018_80B_a66,
     author = {Mathias Lepoutre},
     title = {Bijective {Proof} of the {Rationality} of the {Generating} {Series} of {Higher-Genus} {Maps}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2018},
     volume = {80B},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a66/}
}
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TI  - Bijective Proof of the Rationality of the Generating Series of Higher-Genus Maps
JO  - Séminaire lotharingien de combinatoire
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VL  - 80B
UR  - http://geodesic.mathdoc.fr/item/SLC_2018_80B_a66/
ID  - SLC_2018_80B_a66
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%0 Journal Article
%A Mathias Lepoutre
%T Bijective Proof of the Rationality of the Generating Series of Higher-Genus Maps
%J Séminaire lotharingien de combinatoire
%D 2018
%V 80B
%U http://geodesic.mathdoc.fr/item/SLC_2018_80B_a66/
%F SLC_2018_80B_a66

Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

Bender and Canfield proved in 1991 that the generating series of maps in higher genus is a rational function of the generating series of planar maps. In this paper, we give the first bijective proof of this result. Our approach starts with the introduction of a canonical orientation that enables us to construct a bijection between $4$-valent bicolorable maps and a family of unicellular blossoming maps.