On the minimal generating sets of the Eulerian ideal

TL;DR

This paper characterizes the minimal generating sets of the Eulerian ideal for simple graphs, providing explicit constructions, bounds on degrees, and special cases for bipartite and complete graphs.

Contribution

It introduces a lattice ideal perspective, explicitly constructs minimal generators, and establishes bounds on the maximal generating degree based on graph properties.

Findings

Eulerian ideal is a lattice ideal.

Explicit minimal homogeneous generating set provided.

Maximal generating degree bounded by graph invariants.

Abstract

We study the minimal homogeneous generating sets of the Eulerian ideal associated with a simple graph and its maximal generating degree. We show that the Eulerian ideal is a lattice ideal and use this to give a characterization of binomials belonging to a minimal homogeneous generating set. In this way, we obtain an explicit minimal homogeneous generating set. We find an upper bound for the maximal generating degree in terms of the graph. This invariant is half the number of edges of a largest Eulerian subgraph of even cardinality without even-chords. We show that for bipartite graphs this invariant is the maximal generating degree. In particular, we prove that if the graph is bipartite, the Eulerian ideal is generated in degree $2$ if and only if the graph is chordal. Furthermore, we show that the maximal generating degree is also $2$ when the graph is a complete graph.