-parking functions, acyclic orientations and spanning trees

Given an undirected graph G = ( V , E ) , and a designated vertex q ∈ V , the notion of a G -parking function (with respect to q ) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study the properties of maximum G -parking functions and provide a new bijection between them and the set of spanning trees of G with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the discrete n -cube Q n . We present the article in an expository self-contained form, since we found the combinatorial aspects of G -parking functions somewhat scattered in the literature, typically treated in conjunction with sandpile models and closely related chip-firing games.