on one's honor meaning

[American idiom]
on one's solemn oath; sincerely.
• On my honor, I'll be there on time.
• He promised on his honor that he'd pay me back next week.

[American slang]
Fig. on one's solemn oath; sincerely.
  On my honor, I'll be there on time.
  He promised on his honor that he'd pay me back next week.