The require_once() statement includes and evaluates the specified file during the execution of the script. This is a behavior similar to the require() statement, with the only difference being that if the code from a file has already been included, it will not be included again.
You can use the mysql_pconnect and it may keep the DB connection open. It all depends on how your host is setup.
It is probably best to do the connection on every page. I normally just require() a file that has all the connection stuff in it at the begining. If the connection is already open it makes no difference and if its not it connects.
HalfaBee