If, by the 59th day prior to the election, only one person has been nominated as provided in Section 41305 for any elective office to be filled at that election, or no one has been nominated for that office, the board, in its discretion and by resolution, may order that an election not be held for that office and request the board of supervisors to appoint to that office the person nominated or, where no one has been nominated for that office, to appoint to that office a person who the board of supervisors selects. Upon receipt of a request from the board, the board of supervisors shall make that appointment, and the person appointed shall qualify and take office and serve as if elected at a general election.