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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

1. No two adjacent nodes are assigned the same color.
2. No two edges incident to the same node have the same color.
3. 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 forces $K_4$ to have $6$ colors.

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 of $G$ and so not worthy of separate study?

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

---

          [![NodeEdgeColoring][1]][1]

---

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

1. No two adjacent nodes are assigned the same color.
2. No two edges incident to the same node have the same color.
3. 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 thisThis forces $K_4$ to have 6 $5$ 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 of $G$ and so not worthy of separate study?

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

---

          [![NodeEdgeColoring][1]][1]

---

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

1. No two adjacent nodes are assigned the same color.
2. No two edges incident to the same node have the same color.
3. No edge incident to a node has the same color as that node.

I believe this forces $K_4$ to have $6$ colors.

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 of $G$ and so not worthy of separate study?

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

---

          [![NodeEdgeColoring][1]][1]

---

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

1. No two adjacent nodes are assigned the same color.
2. No two edges incident to the same node have the same color.
3. 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 forces $K_4$ to have $6$ colors.

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 of $G$ and so not worthy of separate study?