Depth of Powers of Edge Ideals of Graphs

Let G be a finite simple graph with n vertices, and let S=K[x_1,…,x_n] be the polynomial ring over a field K. The edge ideal of G is a squarefree monomial ideal of S defined as I(G)=(x_i x_j: {x_i,x_j } is an edge of G ). Edge ideals, first introduced by Villarreal in [10], serve as a bridge between commutative algebra and combinatorics, allowing for algebraic properties of ideals to be explored using combinatorial tools. Since the introduction of this concept the study of these ideals has increasingly drawn the attention of researchers. The most significant aspect lies in correlating the algebraic properties and invariants of these ideals into combinatorial invariants of the underlying graph, and vice versa. In this context, investigating the behaviour of powers of these ideals and understanding their connection to the structure of the underlying graph is of particular importance and interest. One of such problems is exploring the behaviour of the depth function of the ideal. For a graded ideal I in the polynomial ring S over a field, the depth function of the ideal I is defined as f(k)=depth(S/I^k ). The significance of this topic is rooted in a beautiful result by Brodmann [1], which states that the function f stabilizes. Identifying the final value of this function, as well as examining the behaviour of f for smaller values of k (i.e., the behaviour of the initial powers of the ideal) has been studied in numerous papers (see [2,3,4,6,7,8,9,10] and the reference therein). When I is the edge ideal of a graph G, the aim is to find properties of the graph G that correspond to properties of the depth function. Even in this case, the behaviour of the initial powers of the ideal can be complex. Herzog and Hibi [5] classified edge ideals of graphs whose second power has depth zero and showed that such a property for depth can be interpreted purely combinatorial. Specifically, they proved that the second power of the edge ideal of a graph G has depth zero if and only if G is a connected graph with a dominating induced cycle of length 3. In this paper, inspired by the work of Herzog and Hibi we study the third power of edge ideals of graphs and provide a combinatorial classification of edge ideals whose third power has depth zero. In fact, we show that such condition is equivalent to the existence of specific types of dominating induced subgraphs in the graph G. This result particularly demonstrates that if the second power of the edge ideal of a graph has depth zero, then the third power also has depth zero.