Who knows the name of the following coloring of graphs, a proper vertex coloring so that for every vertex its every two neighbors receive different colors?



What's the name of a special vertex coloring
For a graph $G$, the *$t$-th power* of $G$ is the graph $G^t$ with the same vertex set as $G$ and where two vertices are adjacent in $G^t$ if they are connected by a path with at most $t$ edges in $G$. The *distance-$t$ chromatic number* of $G$, often denoted $\chi\_t(G)$, is the chromatic number of $G^t$. As noted by Sam Hopkins in the comments, you are asking about $\chi\_2(G)$. So, your colouring is known as a *distance-$2$ colouring* of $G$. See this [paper](https://arxiv.org/abs/1701.07639) of Kang and Pirot, where this terminology and notation is used.


