2020/04/28 by Chanchal Kumar, Kumar, Chanchal, Gargi Lather +2 · 2 citations
Computer Science · Mathematics · #05E40 #13D02 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2004.13814
openalex publication_date 2020/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be an (oriented) graph on the vertex set V = \ 0, 1,…,n\ with root 0. Postnikov and Shapiro associated a monomial ideal MG in the polynomial ring R = \mathbbK[x1,…,xn] over a field \mathbbK. A subideal MG(k) of MG generated by subsets of \widetildeV=V∖ \0\ of size at most k+1 is called a k-skeleton ideal of the graph G. Many interesting homological and combinatorial properties of 1-skeleton ideal MG(1) are obtained by Dochtermann for certain classes of simple graph G. A finite sequence P=(p1,…,pn) ∈ ℕn is called a spherical G-parking function if the monomial xP = ∏i=1n xipi ∈ MG ∖ MG(n-2). Let \rm sPF(G) be the set of all spherical G-parking functions. In this paper, a combinatorial description for all multigraded Betti numbers of the k-skeleton ideal M_Kn+1(k) of the complete graph Kn+1 on V are given. Also, using DFS burning algorithms of Perkinson-Yang-Yu (for simple graph) and Gaydarov-Hopkins (for multigraph), we give a combinatorial interpretation of spherical G-parking functions for the graph G = Kn+1- \e\ obtained from the complete graph Kn+1 on deleting an edge e. In particular, we showed that |\rm sPF(Kn+1- \e0\ )|= (n-1)n-1 for an edge e0 through the root 0, but |\rm sPF(Kn+1 - \e1\)| = (n-1)n-3(n-2)2 for an edge e1 not through the root.