There must be work on this concept, but I am not finding it through searches, perhaps using the wrong terminology.

Define a

*node-edge coloring*of a graph $G=(V,E)$ to assign an integer color to each node and edge of $G$, such that

- No two adjacent nodes are assigned the same color.
- No two edges incident to the same node have the same color.
- No edge incident to a node has the same color as that node. Or, equivalently, a node's color is distinct from all its incident edges colors.

~~I believe this~~This forces $K_4$ to have ~~6~~ $5$ colors
(Thanks to Fedor Petrov for the coloring.)

. Has this type of coloring been studied? Does it have a name in the literature? Or is it instead just a combination of $G$ and the line graph of $G$ and so not worthy of separate study?Q

total coloringis what you seek? $\endgroup$