On vanishing patterns in $j$-strands of edge ideals

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, $t_{a+b}\le t_a+t_b$, 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 $S\mathrm{/}I$