Comparing symbolic powers of edge ideals of weighted oriented graphs

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 $s \geq 2.$