From semi-total to equitable total colorings, and their induced partitions into perfect and total perfect codes

Independently posed by Behzad and Vizing, the Total Coloring Conjecture asserts that the total chromatic number of a connected simple graph $G$ is either $Δ(G)+1$ or $Δ(G)+2$, where $Δ(G)$ is the largest degree of a vertex of $G$. Deciding whether a cubic graph $G$ has total chromatic number $Δ(G)+1