mysqldump error 2013: lost connection to mysql

To get more debugging information from cron you can specify the following:

0 1 * * * /path_to_directory/file_name.sh > /path_to_directory/file_name.out 2>&1

Also, how big is the database you're backing up?
What commands are you using in mysqldump? (ex:  --opt -C )

#!/bin/sh
#
# mysqlbackup
#
# Script to make daily dumps of MySQL database and RT (if installed)
#
BACKUPDIR=/backup/hq/replicated/mysql

# Backup MySQL
/usr/bin/mysqldump --single-transaction --quick --all-databases --default-character-set=latin1
 > ${BACKUPDIR}/`hostname`-`date +%C%y%m%d`-mysql-all.sql


# Backup RT, if present
if [ -d /usr/local/rt ]; then
    tar cf ${BACKUPDIR}/`hostname`-`date +%C%y%m%d`-rtconfig.tgz -C /usr/local/rt etc
fi

# Clean out old dumps
find ${BACKUPDIR} -ctime +5 -exec rm {} \;

database size is 32107M

Your database is over 30 gigabytes in size? That would definitely take a while to back up. I would again suggest that you examine the other cron scripts to see what runs. That will help you determine what is causing MySQL to stop while your backup is running.

I made an error. The actual database size is 12Gig

The same comment still stands. Your database is 12 GB. Presumably this takes a long time, perhaps hours, to back up. Have you checked your other scheduled tasks to see what may be interfering? I still have not seen you comment on that.

How would I do this with rsync...

BACKUPDIR=/backup/hq/replicated/mysql

# Backup MySQL
/usr/bin/mysqldump --single-transaction --quick --all-databases --default-character-set=latin1 > ${BACKUPDIR}/`hostname`-`date +%C%y%m%d`-mysql
-all.sql

here is the manual for rsync...
http://www.hmug.org/man/1/rsync.php

what you need to do is stop a slave and copy the files directly.
You can also use scp.