  We consider two problems regarding vanishing patterns in the Betti table of
edge ideals $I$ in polynomial algebra $S$. First, we show that the $j$-strand
is connected if $j=3$ (for $j=2$ this is easy and known), and give examples
where the $j$-strand is not connected for any $j>3$. Next, we apply our result
on strand connectivity to establish the subadditivity conjecture for edge
ideals, $t_{a+b}\leq t_a+t_b$, in case $b=2,3$ (the case $b=1$ is known). Here
$t_i$ stands for the maximal shifts in the minimal free $S$-resolution of $S/I$
