This question is related to Realizing the diameter of a finite regular graph

Let $X=(V,E)$ be a finite, connected, regular graph of diameter $D$. Assume that, for every vertex $x\in V$, there exists some vertex $y\in V$ such that $d(x,y)=D$ (btw, does this property has a name in the literature?)

Question: does there exist a permutation $\alpha$ of $V$ such that $d(x,\alpha(x))=D$ for every $x\in V$?

Note that $\alpha$ is NOT required to be a graph automorphism.

Example: let $G$ be a finite group, and let $X$ be a Cayley graph of $G$ wrt some symmetric generating set $S$; use right multiplications by generators to define $X$, so that the distance $d$ is left-invariant. Let $g$ be any element of maximal word length in $G$. Then $\alpha(x)=xg$ (right multiplication by $g$) does the job in the question.

This question is related to Realizing the diameter of a finite regular graph

Let $X=(V,E)$ be a finite, connected, regular graph of diameter $D$. Assume that, for every vertex $x\in V$, there exists some vertex $y\in V$ such that $d(x,y)=D$ (btw, does this property has a name in the literature?)

Question: does there exist a permutation $\alpha$ of $V$ such that $d(x,\alpha(x))=D$ for every $x\in V$?

Note that $\alpha$ is NOT required to be a graph automorphism.

Example: let $G$ be a finite group, and let $X$ be a Cayley graph of $G$ wrt some symmetric generating set $S$; use right multiplications by generators to define $X$, so that the distance $d$ is left-invariant. Let $g$ be any element of maximal word length in $G$. Then $\alpha(x)=xg$ (right multiplication by $g$) does the job in the question.