TOTAL CHROMATIC NUMBER OF GRAPHS OF HIGH DEGREE .2.

We prove Theorem 1: suppose G is a simple graph of order n having DELTA(G) = n - k where k greater-than-or-equal-to 5 and n greater-than-or-equal-to max(13, 3k - 3) . If G contains an independent set of k - 3 vertices, then the TCC (Total Colouring Conjecture) is true. Applying Theorem 1, we also prove that the TCC is true for any simple graph G of order n having DELTA(G) = n - 5. The latter result together with some earlier results confirm that the TCC is true for all simple graphs whose maximum degree is at most four and for all simple graphs of order n having maximum degree at least n - 5.