On the Hilbert depth of the quotient ring of the edge ideal of a star graph

Let S_n = K[x₁, . . . , x_n, y] and I_n = (x₁y, x₂y, . . . , x_ny) ⊂ S_n be the edge ideal of star graph. We prove that hdepth (Sn/In)≥⌈ n2 ⌉+⌊ n ⌋-2 \left( {S_n /I_n } \right) \ge \left\lceil {{n \over 2}} \right\rceil + \left\lfloor {\sqrt n } \right\rfloor - 2 . Also, we show that for any ε > 0, there exists some integer A = A(ε) ≥ 0 such that hdepth (Sn/In)≤⌈ n2 ⌉+⌊ εn ⌋+A−2 (S_n/I_n) \leq \lceil \frac{n}{2} \rceil + \lfloor \varepsilon n \rfloor + A - 2 . We deduce that limn→∞1nhdepth(Sn/In)=12 \lim_{n \to \infty} \frac{1}{n} hdepth(S_n/I_n) = \frac{1}{2} .