2019/09/06 by Neves, Jorge, Pinto, Maria Vaz, Villarreal, Rafael H.
#05C70 #05E40 #13F20 #13P20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1909.02773
We introduce a new class of polynomial ideals associated to a simple graph, G. Let K[EG] be the polynomial ring on the edges of G and K[VG] the polynomial ring on the vertices of G. We associate to G an ideal, I(XG), defined as the preimage of (xi2-xj2 : i,j∈ VG)⊆ K[VG] by the map K[EG]→ K[VG] which sends a variable, te, associated to an edge e=\i,j\, to the product xixj of the variables associated to its vertices. We show that K[EG]/I(XG) is a one-dimensional, Cohen-Macaulay, graded ring, that I(XG) is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field K. We focus on the Castelnuovo-Mumford regularity of I(XG) providing the following sharp upper and lower bounds: μ(G) ≤ reg I(XG) ≤ |VG|-b0(G)+1, where μ(G) is the maximum vertex join number of the graph and b0(G) is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.