Skeleton ideals of certain graphs, standard monomials and spherical parking functions (Q2662332)

Summary: Let \(G\) be a graph on the vertex set \(V = \{ 0, 1,\ldots,n\}\) with root \(0\). \textit{A. Postnikov} and \textit{B. Shapiro} [Trans. Am. Math. Soc. 356, No. 8, 3109--3142 (2004; Zbl 1043.05038)] were the first to consider a monomial ideal \(\mathcal{M}_G\), called the \(G\)-parking function ideal, in the polynomial ring \(R = \mathbb{K}[x_1,\ldots,x_n]\) over a field \(\mathbb{K}\) and explained its connection to the chip-firing game on graphs. The standard monomials of the Artinian quotient \(\frac{R}{\mathcal{M}_G}\) correspond bijectively to \(G\)-parking functions. \textit{A. Dochtermann} [``One-skeleta of \(G\)-parking function ideals: resolutions and standard monomials'', Preprint, \url{arXiv:1708.04712}] introduced and studied skeleton ideals of the graph \(G\), which are subideals of the \(G\)-parking function ideal with an additional parameter \(k\) \((0\leqslant k \leqslant n-1)\). A \(k\)-skeleton ideal \(\mathcal{M}_G^{(k)}\) of the graph \(G\) is generated by monomials corresponding to non-empty subsets of the set of non-root vertices \([n]\) of size at most \(k+1\). Dochtermann [loc. cit.] obtained many interesting homological and combinatorial properties of these skeleton ideals. In this paper, we study the \(k\)-skeleton ideals of graphs and for certain classes of graphs provide explicit formulas and combinatorial interpretation of standard monomials and the Betti numbers.