May I have only my name on the deed, but both my daughter and I on the mortgage?

Technically the lender required both borrowers to be owners if the property at the time the loan was approved and there is typically a clause in the mortgage that says that if the property is transferred to someone other than the borrowers, then the mortgage is due in full. Moreover, the second borrower is still on the hook for the loan, even if you transfer ownership completely to yourself. From a practical point, the lender typically will not default the loan upon transfer as long as it remains current.