Let n≥2 be an integer, and let i∈{0,...,n−1}. An i-th dimension
edge in the n-dimensional hypercube Qn is an edge v1v2 such that
v1,v2 differ just at their i-th entries. The parity of an i-th
dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its
vertex ignoring the i-th entry. We prove that the number of i-th dimension
edges appearing in a given Hamiltonian cycle of Qn with parity zero
coincides with the number of edges with parity one. As an application of this
result it is introduced and explored the conjecture of the inscribed squares in
Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in Qn contains
two opposite edges in a 4-cycle. We prove this conjecture for n≤7, and
for any Hamiltonian cycle containing more than 2n−2 edges in the same
dimension. This bound is finally improved considering the equi-independence
number of Qn−1, which is a concept introduced in this paper for bipartite
graphs