[Repost of same question math stack exchange which got no answers]

I'm looking for literature on the following family of graphs:

Call a regular graph $G=(V,E)$ (of regularity degree $d$) **nice** if there exists a coloring $C:V\to \{ 1,\dots,d \}$ such that each vertex $v$ has precisely one neighbour of each color.

In particular, I'm wondering

**Qustion.** Is the graph on $V=S_n$ with $x\sim y$ iff $y=x\tau$ for some transposition $\tau$ nice?

Here is what I know so far about nice graphs:

By double counting $\{(x,y):x\sim y, C(y)=i\}$ we see that in a nice graph we have $d\mid n = |V|$ and each color appears on $n/d$ vertices. However this property is not sufficient, as a $6$ cycle is $2$ regular but not nice.

An example of a nice graph: $V=\{0,1\}^8$, $x\sim y$ iff they differ in precisely one bit (Hamming distance 1). The coloring of $(x_0,\dots,x_7)$ is $(x_0\cdot 0)\oplus (x_1\cdot 1) \oplus \dots \oplus (x_7\cdot 7)$, with $\oplus$ denoting xor. (Of course $8$ can be replaced with any power of two).