If a vacancy should occur in the office of mayor or councilman, the council shall appoint a person to fill such vacancy until the next general municipal election. If at any municipal election held under section 4 of this Charter, a mayor or the required number of councilmen be not elected by reason of a tie vote among any of the candidates therefor, the council, after the qualifications of the persons, if any, elected thereto at such election, shall appoint one of the persons receiving such tie vote to fill such office as in the case of a vacancy therein. In each case of such vacancy or tie vote, the person so appointed shall hold office, subject to the provisions of the recall, until the next general municipal election at which said election said vacancy shall be filled by the election of an eligible person to serve until the end of the term for which the officer creating such vacancy had been, or, in the case of such tie vote, would have been elected.