Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions

Abstract

Let G be a graph on the vertex set V = { 0 , 1 , … , n } with root 0 . Postnikov and Shapiro were the first to consider a monomial ideal M G , called the G -parking function ideal, in the polynomial ring R = K [ x 1 , … , x n ] over a field K and explained its connection to the chip-firing game on graphs. The standard monomials of the Artinian quotient R M G correspond bijectively to G -parking functions. Dochtermann introduced and studied skeleton ideals of the graph G , which are subideals of the G -parking function ideal with an additional parameter k

( 0 ≤ k ≤ n − 1 ) . A k -skeleton ideal M ( k ) G 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 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.