Minimal Free Resolutions of the G-parking Function Ideal and the Toppling Ideal

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The G-parking function ideal MG of a directed multigraph G is a monomial ideal which encodes some of the combinatorial information of G. It is an initial ideal of the toppling ideal IG, a lattice ideal intimately related to the chip-firing game on a graph. Both ideals were first studied by Cori, Rossin, and Salvy. A minimal free resolution for MG was given by Postnikov and Shaprio in the case when G is saturated, i.e., whenever there is at least one edge (u,v) for every ordered pair of distinct vertices u and v. They also raised the problem of an explicit description of the minimal free resolution in the general case. In this paper, we give a minimal free resolution of MG for any undirected multigraph G, as well as for a family of related ideals including the toppling ideal IG. This settles a conjecture of Manjunath and Sturmfels, as well as a conjecture of Perkinson and Wilmes.