Centre for Mathematics, University of Coimbra

<Reference List>
Type:	Preprint
National /International:	International
Title:	Eulerian ideals
Publication Date:	2020-11-09
Authors:	- Jorge Sentieiro Neves
Abstract:	Let G be a simple graph and I(X_G) = ϕ⁻¹(x_i² − x_j² : i, j ∈ V_G), where ϕ: K[E_G] → K[V_G] is the homomorphism that sends an edge to the product of its vertices. The ideal I(X_G) is Cohen–Macaulay, one-dimensional and binomial. If G is bipartite, it is known that the Castelnuovo–Mumford regularity of I(X_G) is equal to the maximum cardinality of a set of edges having no more than half of the edges of any Eulerian subgraph of G. Here, with respect to the grevlex order associated to an ordering of the edge set of G, we describe a Grobner basis for I(X_G), and we characterize the standard monomials of the ideal (I(X_G),t_e) in terms of even sets of vertices marked with a parity. Using these results, we classify the case of I(X_G) Gorenstein; we give a combinatorial interpretation of the degree of I(X_G), via the set of even sets of vertices of G; and we show that the Castelnuovo–Mumford regularity of I(X_G), for any graph, is the maximum cardinality of a set of edges having no more than half of the edges of any even Eulerian subgraph of G or, equivalently, the maximum cardinality of a minimum fixed parity T-join.
Institution:	DMUC 20-31
Online version:	http://www.mat.uc.pt...prints/eng_2020.html
Download:	Not available