  The $G$-parking function ideal $M_G$ 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 $I_G$, 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 $M_G$ 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 $M_G$ for any undirected multigraph $G$, as well as for a family
of related ideals including the toppling ideal $I_G$. This settles a conjecture
of Manjunath and Sturmfels, as well as a conjecture of Perkinson and Wilmes.
