Degree and regularity of Eulerian ideals of hypergraphs

TL;DR

This paper introduces the Eulerian ideal of hypergraphs, explores its algebraic properties like degree and regularity using combinatorial tools, and provides explicit formulas for special hypergraph classes.

Contribution

It defines the Eulerian ideal for hypergraphs, links its regularity to parity joins, and computes degree and regularity for specific hypergraph types.

Findings

Regularity equals the maximum size of a parity join.

Degree involves counting sets admitting a T-join.

Explicit formulas for complete k-partite and rank 3 hypergraphs.

Abstract

We define the Eulerian ideal of a $k$-uniform hypergraph and study its degree and Castelnuovo--Mumford regularity. The main tool is a Gr\"obner basis of the ideal obtained combinatorially from the hypergraph. We define the notion of parity join in a hypergraph and show that the regularity of the Eulerian ideal is equal to the maximum cardinality of such a set of edges. The formula for the degree involves the cardinality of the set of sets of vertices, $T$, that admit a $T$-join. We compute the degree and regularity explicity in the cases of a complete $k$-partite hypergraph and a complete hypergraph of rank $3$.