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, depth R / I ≥ 1 and depth R / I t = 0 for t ≫ 1 . This paper studies the problem when depth R / 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 depth R / I (t) = 1 for t ≫ 1 and show that the condition depth R / I (t) = 1 is persistent, where I (t) denotes the t-th symbolic powers of I. [ABSTRACT FROM AUTHOR]