One-skeleta of $G$-parking function ideals: resolutions and standard monomials

Given a graph $G$, the $G$-parking function ideal $M_G$ is an artinian monomial ideal in the polynomial ring $S$ with the property that a linear basis for $S/M_G$ is provided by the set of $G$-parking functions. It follows that the dimension of $S/M_G$ is given by the number of spanning trees of $G$, which by the Matrix Tree Theorem is equal to the determinant of the reduced Laplacian of $G$. The ideals $M_G$ and related algebras were introduced by Postnikov and Shapiro where they studied their Hilbert functions and homological properties. The author and Sanyal showed that a minimal resolution of $M_G$ can be constructed from the graphical hyperplane arrangement associated to $G$, providing a combinatorial interpretation of the Betti numbers. Motivated by constructions in the theory of chip-firing on graphs, we study certain `skeleton ideals $M_G^{(k)} subset M_G$ generated by subsets of vertices of $G$ of size at most $k+1$. Here we focus our attention on the case $k=1$, the $1$-skeleton of the $G$-parking functions ideals. We consider standard monomials of $M_G^{(1)}$ and provide a combinatorial interpretation for the dimension of $S/M_G^{(1)}$ in terms of the signless Laplacian for the case $G = K_{n+1}$ is the complete graph. Our main study concerns homological properties of these ideals. We study resolutions of $M_G^{(1)}$ and show that for a certain class of graphs minimal resolution is supported on decompositions of Euclidean space coming from the theory of tropical hyperplane arrangements. This leads to combinatorial interpretations of the Betti numbers of these ideals.