From Longman Dictionary of Contemporary Englishon your honouron your honoura) PROMISEif you swear on your honour to do something, you promise very seriously to do it b) old-fashionedTRUST if you are on your honour to do something, you are being trusted to do it → honour
Examples from the Corpus
on your honour• But I have placed the boy on his honour.• Auguste would take it as a reflection on his honour if they were not, despite the unseasonable time of year.• They are a wild, free-spirited people, quick-tempered as Elves go, and swift to avenge any stain on their honour. 