There must be work on this concept, but I am not finding it through
searches, perhaps using the wrong terminology.
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[![NodeEdgeColoring][1]][1]
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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
<ol>
<li>No two adjacent nodes are assigned the same color.</li>
<li>No two edges incident to the same node have the same color.</li>
<li>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.</li>
</ol>
<strike>I believe this</strike>This forces $K_4$ to have <strike>6</strike> $5$ colors
(Thanks to Fedor Petrov for the coloring.)

> ***Q***. 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](http://mathworld.wolfram.com/LineGraph.html) of $G$
and so not worthy of separate study?


  [1]: https://i.sstatic.net/ortnc.jpg