On the Stanley depth of powers of edge ideals

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Let $G$ be a graph with $n$ vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $p$ is the number of its bipartite connected components. We prove that for every positive integer $k$, the inequalities ${\rm sdepth}(I^k/I^{k+1})\geq p$ and ${\rm sdepth}(S/I^k)\geq p$ hold. As a consequence, we conclude that $S/I^k$ satisfies the Stanley's inequality for every integer $k\geq n-1$. Also, it follows that $I^k/I^{k+1}$ satisfies the Stanley's inequality for every integer $k\gg 0$. Furthermore, we prove that if (i) $G$ is a non-bipartite graph, or (ii) at least one of the connected components of $G$ is a tree with at least one edge, then $I^k$ satisfies the Stanley's inequality for every integer $k\geq n-1$. Moreover, we verify a conjecture of the author in special cases.

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