You could establish a persistent connection using
mysql_pconnect() rather than mysql_connect()
mysql_pconnect() acts very much like mysql_connect() with two major differences.
First, when connecting, the function would first try to find a (persistent) link that's already open with the same host, username and password. If one is found, an identifier for it will be returned instead of opening a new connection.
Second, the connection to the mysql server will not be closed when the execution of the script ends. Instead, the link will remain open for future use (mysql_close() will not close links established by mysql_pconnect()).
Normally you do NOT want to use mysql_pconnect. This function is designed for environments which have a high overhead to connecting to the database. In a typical MySQL / Apache / PHP environment, Apache will create many child processes which lie in idle waiting for a web request to be assigned to them. Each of these child processes will open and hold its own MySQL connection. So if you have a MySQL server which has a limit of 50 connections, but Apache keeps more than 50 child processes running, each of these child processes can hold a connection to your MySQL server, even while they are idle (idle httpd child processes don't lend their MySQL connection to other httpd children, they hold their own). So even if you only have a few pages which actually connect to MySQL on a busy site, you can run out of connections, with all of them not actually being used.
In general, use mysql_connect() for connecting to MySQL unless that connection takes a long time to establish.