Total chromatic number of graphs of high degree, II
Hian Poh Yap, Kim Ho Chew · Journal of the Australian Mathematical Society Series A Pure Mathematics and Statistics · 1992
Abstract We prove Theorem 1: suppose G is a simple graph of order n having Δ(G) = n − k where k ≥ 5 and n ≥ 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 Δ(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.