Eulerian ideals

Abstract

Let G be a simple graph and , where is the homomorphism that sends an edge to the product of its vertices. The ideal is Cohen–Macaulay, one-dimensional and binomial. If G is bipartite, it is known that the Castelnuovo–Mumford regularity of 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 , and we characterize the standard monomials of the ideal in terms of even sets of vertices marked with a parity. Using these results, we give a combinatorial interpretation of the degree of , via the set of even sets of vertices of G; and we show that the Castelnuovo–Mumford regularity of , 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.