It doesn't HAVE to have the entire domain...for MANY email servers. Some are really, really picky. Once a client of mine has been blacklisted, often I find the only way to make EVERYone happy (read: "Certain obnoxious ISPs, especially some cable TV companies") is to use the fully qualified domain name.

So my advice--which you are perfectly entitled to not follow if you don't want to--is to use the FQDN.