Eulerian ideals

Abstract

Let G be a simple graph and $I(X_G)={\phi }^{-1}(x_i^2-x_j^2:i,j\in V_G)$, where $\phi :K[E_G]\to K[V_G]$ 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 Gröbner 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 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.

Keywords:

• Binomial ideal
• Castelnuovo–Mumford regularity
• T-joins

2020 Mathematics Subject Classification:

• 13A02
• 13H10
• 13P25
• 05E40
• 05C70
• 05C45

Acknowledgments

The author thanks Jens Vygen and András Sebő for a helpful discussion on the subject of T-joins. The relation between the notions of parity joins and of fixed parity T-joins in Lemma 4.12 was pointed out by András Sebő.

Additional information

Funding

This work was partially supported by the Centre for Mathematics of the University of Coimbra – UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.