A family of bijections between G-parking functions and spanning trees

Abstract

For a directed graph G on vertices {0,1,...,n}, a G-parking function is an n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty subset U of {1,...,n}, there exists a vertex j in U for which there are more than b_j edges going from j to G-U. We construct a family of bijective maps between the set P_G of G-parking functions and the set T_G of spanning trees of G rooted at 0, thus providing a combinatorial proof of |P_G| = |T_G|.