Is it safe to delete mysql-bin files?
Please do not just delete them in the OS.
You need to let mysqld do that for you. Here is how mysqld manages it:
The file mysql-bin.[index] keeps a list of all binary logs mysqld has generated and auto-rotated. The mechanisms for cleaning out the binlogs in conjunction with mysql-bin.[index] are:
PURGE BINARY LOGS TO 'binlogname';
PURGE BINARY LOGS BEFORE 'datetimestamp';

These will clear all binary logs before the binlog or timestamp you just specified.
For example, if you run
PURGE BINARY LOGS TO 'mysql-bin.000223';

this will erase all binary logs before mysql-bin.000223.
If you run
PURGE BINARY LOGS BEFORE DATE(NOW() - INTERVAL 3 DAY) + INTERVAL 0 SECOND;

this will erase all binary logs before midnight 3 days ago.
If you want to have binlog rotated away automatically and keep 3 days woth, simply set this:
mysql&gt; SET GLOBAL expire_logs_days = 3;

then add this to /etc/my.cnf
[mysqld]
expire_logs_days=3

and mysqld will delete them logs for you
SHOW SLAVE STATUS\G
This is critical. When you run SHOW SLAVE STATUS\G, you will see two binary logs from the Master:

Master_Log_File
Relay_Master_Log_File

When replication has little or no lag these are usually the same value. When there is a lot of replication lag, these values are different. Just to make it simple, choose whatever Relay_Master_Log_File is, and go back to the Master and run
PURGE BINARY LOGS TO 'Whatever Relay_Master_Log_File Is';

That way, replication is not interrupted.