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We investigate the integrability of the discrete non-linear equationgoverning the dependence on geodesic distance of planar graphs with innervertices of even valences. This equation follows from a bijection betweengraphs and blossom trees and is expressed in terms of generating functions forrandom walks. We construct explicitly an infinite set of conserved quantitiesfor this equation, also involving suitable combinations of random walkgenerating functions. The proof of their conservation, i.e. their eventualindependence on the geodesic distance, relies on the connection between randomwalks and heaps of dimers. The values of the conserved quantities areidentified with generating functions for graphs with fixed numbers of externallegs. Alternative equivalent choices for the set of conserved quantities arealso discussed and some applications are presented.Comment: 38 pages, 15 figures, uses epsf, lanlmac and hyperbasic

Integrability of graph combinatorics via random walks and heaps of dimers