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, \({\text {depth}}R/I \ge 1\) and \({\text {depth}}R/I^t = 0\) for \(t \gg 1\). This paper studies the problem when \({\text {depth}}R/I^t = 1\) for some \(t \ge 1\) and whether the depth function is non-increasing thereafter. Furthermore, we are able to give a simple combinatorial criterion for \({\text {depth}}R/I^{(t)} = 1\) for \(t \gg 1\) and show that the condition \({\text {depth}}R/I^{(t)} = 1\) is persistent, where \(I^{(t)}\) denotes the t-th symbolic powers of I.