A note on the minimum total coloring of planar graphs

Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is also called total coloring. We consider a planar graph G with maximum degree Delta(G) a parts per thousand yen 8, and proved that if G contains no adjacent i, j-cycles with two chords for some i, j a {5, 6, 7}, then G is total-(Delta + 1)-colorable.