\hsize=6 true in
{\bf\noindent Gilles Schaeffer }\smallskip
{\bf\noindent Bijective Census and Random Generation of Eulerian Planar Maps with
       Prescribed Vertex Degrees }\vskip.5cm
\noindent 
Abstract: We give a bijection between Eulerian planar maps with
prescribed vertex degrees, and some plane trees that we call
balanced Eulerian trees.} To enumerate the latter, we
introduce conjugation classes of planted plane trees. In particular,
the result answers a question of Bender and Canfield  and allows uniform random
generation of Eulerian planar maps with restricted vertex
degrees. Using a well known correspondence between 4-regular planar
maps with n vertices and planar maps with n edges we obtain an
algorithm to generate uniformly such maps with complexity O(n). Our
bijection is also refined to give a combinatorial interpretation of a
parameterization of Arqu\`es  of the generating
function of planar maps with respect to vertices and faces. 

\bye