Comparing symbolic powers of edge ideals of weighted oriented graphs

TL;DR

This paper investigates when symbolic powers of edge ideals of weighted oriented graphs equal their ordinary powers, providing characterizations for specific graph classes and conditions involving odd cycles.

Contribution

It establishes necessary and sufficient conditions for the equality of symbolic and ordinary powers in weighted oriented graphs, especially those with odd cycles and star graphs.

Findings

Equality holds for weighted oriented star graphs for all powers s ≥ 2.

Characterization of weighted naturally oriented unicyclic graphs with unique odd cycles.

Similar behavior of symbolic powers in graphs with replaced vertex weights.

Abstract

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. If $D$ contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of a weighted oriented graph having each edge in some induced odd cycle of it. We characterize the weighted naturally oriented unicyclic graphs with unique odd cycles and weighted naturally oriented even cycles for the equality of ordinary and symbolic powers of their edge ideals. Let $ D^{\prime} $ be the weighted oriented graph obtained from $D$ after replacing the weights of vertices with non-trivial weights which are sinks, by trivial weights. We show that the symbolic powers of $I(D)$ and $I(D^{\prime})$ behave in a similar way. Finally, if $D$ is any weighted oriented star graph, we show that $ {I(D)}^{(s)} = {I(D)}^s $ for all $s \geq 2.$