No, dots are NOT necessary. Actually you do not even need to supply a domain or a top level domain because mails then default to the default system which is usually localhost.
But even for routed mail there doesn’t need to be a dot.
There is still valid Bang-Adressing for UUCP routed emails:
!bigsite!foovax!barbox!me
This is a valid email which basically means “send my email to bigsite, from there to foovax, then to barbox, to the user me.”
And if you are in a playful mood - mix FQDN and BANG addressing…
A couple of years ago I made Hotmail crash by sending a mail to googlemail.de!hotmail.com!googlemail.com!hotmail.de!googlemail.ca!hotmail.ca!googlemail.fr!hotmail.fr!.. [repeated it for 32kByte] …!myuseraccount - their server literally crashed completely all over the world for like 15 minutes. I am so proud of myself but then it was their fault for not complying to RfC822.