Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. In this paper, we prove that $I(D)$ satisfies the Ratliff (strong persistence) property in the following three cases: (i) $D$ has an outward leaf; (ii) $D$ has an inward leaf $(u,v)\in E(D)$, where $v$ is a sink vertex; (iii) $D$ has an inward leaf $(u,v)\in E(D)$ with $w(v)=1$. We further show that $(I(D)^2:I(D))=I(D)$ if $D$ contains a vertex with in-degree less than or equal to 1, and $(I(D)^3:I(D))=I(D)^2$ when $D$ is either a weighted oriented cycle, or a tree. Finally, if $D$ contains no source vertex, then any associated prime of $I(D)^k$, other than the irrelevant maximal ideal, is also an associated prime of $I(D)^{k+1}$. In addition, if $D$ contains a vertex of in-degree one and all the vertices of $D$ have non-trivial weights, we show that the persistence property holds.
@article{10_37236_13223,
author = {Arindam Banerjee and Kanoy Kumar Das and Pritam Roy},
title = {Ratliff property of edge ideals of weighted oriented graphs},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {4},
doi = {10.37236/13223},
zbl = {8120095},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13223/}
}
TY - JOUR
AU - Arindam Banerjee
AU - Kanoy Kumar Das
AU - Pritam Roy
TI - Ratliff property of edge ideals of weighted oriented graphs
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/13223/
DO - 10.37236/13223
ID - 10_37236_13223
ER -
%0 Journal Article
%A Arindam Banerjee
%A Kanoy Kumar Das
%A Pritam Roy
%T Ratliff property of edge ideals of weighted oriented graphs
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/13223/
%R 10.37236/13223
%F 10_37236_13223
Arindam Banerjee; Kanoy Kumar Das; Pritam Roy. Ratliff property of edge ideals of weighted oriented graphs. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13223
