How can I prove that revolving door graph $RD^d$ is $d$-regular graph?
$RD^d$ is bipartite graph in which the vertices are $(d-1)$-element subsets and $d$-element subsets of a $(2d-1)$-element ground set. Two vertices are adjacent if one of the corresponding sets is a subset of the other.
First focus on a $(d-1)$-element subset $X$. In how many ways can you choose another element from the $2d-1$ elements to enlarge $X$ to size $d$?
Then focus on a $d$-element subset $Y$. In how many ways can you remove an element from $Y$ to shrink it to size $d-1$.
Conclusion?