Eulerian ideals

TL;DR

This paper studies the algebraic properties of Eulerian ideals associated with graphs, providing a Gröbner basis, combinatorial interpretations, and regularity characterizations related to Eulerian subgraphs and parity conditions.

Contribution

It introduces a Gröbner basis for Eulerian ideals, characterizes standard monomials with parity conditions, and links algebraic invariants to combinatorial structures in graphs.

Findings

Gröbner basis for the ideal $I(X_G)$ described under grevlex order.

Standard monomials characterized by even sets of vertices with parity.

Regularity equals the maximum size of edge sets avoiding half of any even Eulerian subgraph.

Abstract

Let $G$ be a simple graph and $I(X_G)=\varphi^{-1}(x_i^2-x_j^2 : i,j\in V_G)$, where $\varphi \colon K[E_G]\to K[V_G]$ 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 Gr\"obner 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 \emph{even} Eulerian subgraph of $G$ or, equivalently, the maximum cardinality of a minimum fixed parity $T$-join.