Dic: on someone's watch, on the watch

on someone's watch:
while someone is on duty; while someone is supposed to be in charge of a situation.

• I am not responsible since it didn't happen on my watch.
• I guess I have to bear the blame since it happened on my watch.

cf.

on the watch (for someone or something):
alert and watching for someone or something.

• Please stay on the watch for trouble.
• I'm always on the watch for Ann. I want to know when she's around.

... McGraw-Hill Dictionary of American Idioms and Phrasal Verbs.

http://idioms.thefreedictionary.com/watch

• Environmentalists will be on the watch for damage to wildlife.
• = on the lookout

.... Cobuild