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

Authors:
Denis Chebikin;Pavlo Pylyavskyy
Affiliations:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA
Venue:
Journal of Combinatorial Theory Series A
Year:
2005

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Abstract

For a directed graph G on vertices {0, 1, ..., n}, a G-parking function is an n-tuple (b1,...,bn) of non-negative integers such that, for every non-empty subset U ⊆ {1,...,n}, there exists a vertex j ∈ U for which there are more than bj edges going from j to G - U. We construct a family of bijective maps between the set PG of G-parking functions and the set JG of spanning trees of G rooted at 0, thus providing a combinatorial proof of |PG| = |JG|.