Symbolic depth functions of edge ideals via local graph structure

We study the depth of symbolic powers of edge ideals, focusing on local properties of graphs. First, we give several sufficient conditions for the symbolic depth function to be non-increasing, using local properties of graphs and local cohomology. Next, we give a counterexample to a conjecture of Minh, T.N. Trung, and Vu concerning the stable value of the symbolic depth function, where the difference between the conjectured value and the actual value can be arbitrary large. Finally, we study a conjecture of Muta stating the depth of a symbolic power of an edge ideal is at least that of the corresponding ordinary power. We prove it for graphs whose connected components are unicyclic or bipartite.