TITLE: Graph with total chromatic number $\chi''(G)=\chi'(G)+\chi(G)$
QUESTION [5 upvotes]: 1)I try to show that the graphs with total chromatic number $\chi''(G)=\chi'(G)+\chi(G)$ are exactly the bipartite graphs.
Definition https://en.wikipedia.org/wiki/Total_coloring
After the answer of Smylic
2)If $\chi''(G)=\chi'(G)+\chi(G)$ holds then the graph should be bipartite

REPLY [3 votes]: This is not true, because $\chi''(C_6) = 3 \ne 4 = 2 + 2 = \chi'(C_6) + \chi(C_6)$.