Decreasing behavior of the depth functions of edge ideals

Let I be the edge ideal of a connected non-bipartite graph and R the base polynomial ring. Then, depthR/I ≥ 1 and depthR/I^t = 0 for t ≫ 1 . This paper studies the problem when depthR/I^t = 1 for some t ≥ 1 and whether the depth function is non-increasing thereafter. Furthermore, we are able to give a simple combinatorial criterion for depthR/I^(t) = 1 for t ≫ 1 and show that the condition depthR/I^(t) = 1 is persistent, where I^(t) denotes the t -th symbolic powers of I .