We extend Schaeffer's bijection between rooted quadrangulations andwell-labeled trees to the general case of Eulerian planar maps with prescribedface valences, to obtain a bijection with a new class of labeled trees, whichwe call mobiles. Our bijection covers all the classes of maps previouslyenumerated by either the two-matrix model used by physicists or by thebijection with blossom trees used by combinatorists. Our bijection reduces theenumeration of maps to that, much simpler, of mobiles and moreover keeps trackof the geodesic distance within the initial maps via the mobiles' labels.Generating functions for mobiles are shown to obey systems of algebraicrecursion relations.
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