2012/10/29 by Madhusudan Manjunath, Manjunath, Madhusudan, Frank-Olaf Schreyer +4
Computer Science · Mathematics · #13D02 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Limits and Structures in Graph Theory #Topological and Geometric Data Analysis #math.AC #math.CO #msc:13D02
paper · pdf · doi:10.48550/arxiv.1210.7569
22 pages, 3 figures; v3: minor changes
openalex publication_date 2012/10/29 · arxiv created 2012/12/24 · arxiv updated 2012/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.