Coloring where all colors are present in closed neighborhood of every vertex I am interested in (proper) vertex colorings of graphs with the following condition:
for every vertex $v$ in the graph, all colors should be present in the closed neighborhood of $v$.
Is this studied in the literature? The closest I found is $r$-dynamic $k$-coloring. For $k$-regular graphs, a ($k-1$)-dynamic $k$-coloring satisfies the above condition. But, I am also interested in a coloring that use less than $k$ colors and satisfies the above condition. Currently, I am more interested in such coloring of regular graphs.
If there is a paper dealing with ($k-1$)-dynamic $k$-colorings of $k$-regular graphs, I would like to read it (I saw a result on ($k+1$)-dynamic $k$-colorings in a paper).
Thank you.
Related Colouring Variants:-

$r$-dynamic $k$-coloring (also called $r$-hued $k$-coloring)
An $r$-dynamic $k$-coloring of $G$ is a (proper) $k$-coloring of $G$ such that every vertex $v$ in $G$ has neighbors in at least $\min\{deg(v),r\}$ different color classes.
b-coloring
A b-coloring is a (proper) coloring such that every color class $V_i$ contains a vertex $u_i$ which has neighbors in all other color classes.
In the coloring variant I am looking for, every vertex has neighbors in all other color classes.

Definition:-
For a vertex $v$ of $G$, the closed neighborhood of $G$, denoted by $N[v]$, is $v$ together with its set of neighbors.
i.e., $N[v]=\{v\}\cup\{u : uv\in E(G)\}$.
I would call this a polychromatic coloring of the closed neighborhood hypergraph.
I don't think this has been studied before for general graphs.
Here is a paper studying the question when edges are colored:
Béla Bollobás, David Pritchard, Thomas Rothvoß, Alex Scott: Cover-Decomposition and Polychromatic Numbers
And here is one that studies the conflict-free coloring problem for your hypergraph:
Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, Rogers Mathew: Conflict-free coloring on closed neighborhoods of bounded degree graphs
If you replace closed neighborhood with geometric range spaces, then you can find a lot more literature, most notably my own amazing papers.