A. The title to all school property, both real and personal, within a school division shall be vested in the school board, except that by mutual consent of the school board of a school division composed solely of part or all of a city and the governing body of the city, the title to property may vest in the city.
B. The official care and authority of a school board shall cover all territory included in the geographical boundaries of the school division and all school property located without and contiguous to the boundaries of such school division when the title to such property is vested either in the school board or a city; provided, however, that school property lying without the corporate limits of a city but not adjacent thereto on January 1, 1968, shall be subject to the official care and authority of the school board of such city or the city.