  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.
