Total coloring of planar graphs without 6-cycles

A image-image of a graph image is a coloring of image using image colors such that no two adjacent or incident elements receive the same color. The total chromatic number image is the smallest integer image such that image has a total image-coloring. Let image be a planar graph with maximum degree image and without 6-cycles. In this paper, it is proved that image if image and image contains no 4-cycles, or image and image contains no 5-cycles.