(this question is about trying to use some combinatorics to simplify an algorithm and save memory)
Let $K_{2n+1}$ be a complete undirected graph on $2n+1$ vertices. I would like to generate a Eulerian circuit of this graph (visit each edge exactly once).
One solution is to run the DFS-based algorithm that can find a Eulerian circuit in any Eulerian graph (a graph with all vertices of even degree).
However, I'm hoping it's possible to use the nice structure of $K_{2n+1}$ to avoid constructing the graph and running the algorithm. Is there a way to construct one such circuit in O(n) time with O(1) memory?
Motivation: The motivation behind this is an algorithm which receives a list L and needs to calculate $B(A(x), A(y))$ for each unordered pair ${x,y}$ of distinct elements of L (I'm only considering unordered pairs since $B(u,w)=B(w,u)$). The result of operation A is very large and takes almost half of the available memory. Such a circuit will allow me to minimize the number of calls to operation $A$.
Note: This is a sort of "undirected" version of generating a de-Bruijn sequence. I once heard it's possible to generate a de-Bruijn sequence with constant memory, but I don't know how to do that.