On the subadditivity condition of edge ideal

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