Abstract
The G G -parking function ideal M G M_G of a directed multigraph G G is a monomial ideal which encodes some of the combinatorial information of G G . It is an initial ideal of the toppling ideal I G 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 M_G was given by Postnikov and Shapiro in the case when G G is saturated, i.e., whenever there is at least one edge ( u , v ) (u,v) for every ordered pair of distinct vertices u u and v 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 M_G for any undirected multigraph G G , as well as for a family of related ideals including the toppling ideal I G I_G . This settles a conjecture of Manjunath and Sturmfels, as well as a conjecture of Perkinson and Wilmes.
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