The total chromatic number χT(G) of a graph G is the minimum number of colours needed to colour the edges and the vertices of G so that incident or adjacent elements have distinct colours. We show that if G is a regular graph of even order and δ(G)≥23|V(G)|+236, thenχT(G)≤Δ(G)+2. © 2005 Elsevier B.V. All rights reserved.
Xie, D., & He, Z. (2005). The total chromatic number of regular graphs of even order and high degree. Discrete Mathematics, 300(1–3), 196–212. https://doi.org/10.1016/j.disc.2005.04.016