On the subadditivity condition of edge ideal

Let $S=K[x_1,\ldots,x_n]$, where $K$ is a field, and $t_i(S/I)$ denotes the maximal shift in the minimal graded free $S$-resolution of the graded algebra $S/I$ at degree $i$, where $I$ is an edge ideal. In this paper, we prove that if $t_b(S/I)\geq \lceil \frac{3b}{2} \rceil$ for some $b\geq 0$, then the subadditivity condition $t_{a+b}(S/I)\leq t_a(S/I)+t_b(S/I)$ holds for all $a\geq 0$. In addition, we prove that $t_{a+4}(S/I)\leq t_a(S/I)+t_4(S/I)$ for all $a\geq 0$ (the case $b=0,1,2,3$ is known). We conclude that if the projective dimension of $S/I$ is at most $9$, then $I$ satisfies the subadditivity condition.