This is followup question to Properties of Hamilton cycles in hypercubes
a necessary conditon that in a Hamilton cycle in a hypercube of dimension $n$ all maximal sets of mutually parallel edges can have equal cardinality is $n=2^k$ and $k=1$ proves the existence of such cycles, leading to the
Questions:
- is $n=2^k$ also sufficient for the existence of such Hamilton cycles in hypergraphs, whose maximal sets of mutually parallel edges have equal cardinality?
- how many such Hamilton cycles can exist in hypercubes of dimension $n=2^k$, that are not equivalent under automorphisms of the hypercube?
It would also be interesting to know if the Gray codes that are related to those "balanced" Hamilton cycles have any desirable properties.
