We define the Eulerian ideal of a $k$-uniform hypergraph and study its degree and Castelnuovo-Mumford regularity. The main tool is a Gröbner 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 explicitly in the cases of a complete $k$-partite hypergraph and a complete hypergraph of rank three.
@article{10_37236_11180,
author = {Jorge Neves and Gon\c{c}alo Varej\~ao},
title = {Degree and regularity of {Eulerian} ideals of hypergraphs},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {4},
doi = {10.37236/11180},
zbl = {1506.13039},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11180/}
}
TY - JOUR
AU - Jorge Neves
AU - Gonçalo Varejão
TI - Degree and regularity of Eulerian ideals of hypergraphs
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/11180/
DO - 10.37236/11180
ID - 10_37236_11180
ER -
%0 Journal Article
%A Jorge Neves
%A Gonçalo Varejão
%T Degree and regularity of Eulerian ideals of hypergraphs
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/11180/
%R 10.37236/11180
%F 10_37236_11180
Jorge Neves; Gonçalo Varejão. Degree and regularity of Eulerian ideals of hypergraphs. The electronic journal of combinatorics, Tome 29 (2022) no. 4. doi: 10.37236/11180
