Comparisons between ordinary and symbolic powers of edge ideals with respect to regularity and depth

In this paper, we investigate the difference between ordinary and symbolic powers of edge ideals on the Castelnuovo-Mumford regularity and the depth. As a main theorem, we prove that the regularities of ordinary and symbolic powers coincide for edge ideals of simplicial graphs, as a partial result of Minh’s conjecture. For the depth, we first show that, for each $k\ge 2$, the inequality $\mathrm{depth}S\mathrm{/}{I}^{(k)}\ge \mathrm{depth}S\mathrm{/}{I}^k$ does not hold for squarefree monomial ideals in general. On the other hand, we prove that it holds for edge ideals when $k=2,3$. We also study the symbolic-ordinary discrepancy module $I(G{)}^{(k)}\mathrm{/}I(G{)}^k$ of the edge ideal $I(G)$ of a graph $G$ and give a graph-theoretic formula for its Krull dimension in terms of induced odd cycles of $G$, thereby answering a question and a problem posed by Ha and Minh.