The city commission shall further have the right by ordinance to adopt and prescribe rules and regulations for the handling and disposition of all garbage, trash and rubbish within the City of Texas City and shall further have the right to prescribe that the city alone shall remove all such garbage, trash and rubbish, and shall have the right to fix charges and compensation to be charged by the city for the removal of same, providing rules and regulations for the collection thereof, making the same a personal obligation of the person for whom the same is hauled and disposed of, and providing penalties against all other persons hauling such garbage, trash, or rubbish, and providing for a full and adequate system of covering the City of Texas City or such portions thereof as the city commission may determine for the removal and disposition of all such garbage, trash, and rubbish.