asymptotic behaviour of associated primes of monomial ideals with combinatorial applications

let  $r$ be a commutative noetherian ring and $i$ be an ideal of $r$. we say that $i$ satisfies the persistence property if  $mathrm{ass}_r(r/i^k)subseteq mathrm{ass}_r(r/i^{k+1})$ for all positive integers $kgeq 1$, which $mathrm{ass}_r(r/i)$ denotes the set of associated prime ideals of $i$. in this paper, we introduce a class of square-free monomial ideals in the polynomial ring  $r=k[x_1,ldots,x_n]$ over field $k$ which are associated to  unrooted trees  such that if  $g$ is a unrooted tree and $i_t(g)$ is the ideal generated by the paths of $g$ of length $t$, then $j_t(g):=i_t(g)^vee$, where $i^vee$ denotes the alexander dual of $i$, satisfies the persistence property. we also present a class of graphs such that the path ideals generated by paths of length two satisfy the persistence property. we conclude  this paper by giving a criterion for normally torsion-freeness of monomial ideals.