Total chromatic number $(\chi '')$ is defined as the total number of colours needed to colour edges and vertices of a graph simultaneously with no adjacent vertices having the same colour and with no two incident edges having the same colour.
List colouring conjecture says that, the chromatic number of a line graph is equal to the choice number of the line graph.
I want to show that, if the list colouring conjecture is right, it implies that:
$$ \chi''(G) \leq \Delta(G) + 3$$
For a graph $G$ where $\Delta(G)$ is the maximum degree of any vertex in the graph $G$.
Any help would be appreciated.