Let $S$ be the set of all $k$-element subsets of some given universe $M$. My question is as follows: Is it possible to enumerate (without repetition) the elements of $S$ such that each pair of consecutive sets differ in 2 elements.
I've been thinking about the following approaches, but so far haven't made much progress:
We can think of the sets in $S$ as binary vectors of length |M|, where the vector for set $A$ is 1 at index $i$ iff the $i$-th element of $M$ is in $A$. We can form a graph by using these vectors as vertices and where there is an edge between $2$ vertices if their Hamming distance is exactly $2$. Now the goal becomes finding a Hamiltonian path in this graph.