The connection between a man and woman, man and man, or woman and woman that is very strong, such as the Verizon Network Connection which is very reliable and avaliable in almost every city/state. So when two people share a strong connection, it is referred to as "The Verizon Connection"
Dude: Man, I just brokeup with my girl.
You: Why man?!
Dude: I'm not sure; it didn't feel right. No connection.
You: Man, too bad you guys didn't have the Verizon Connection!