  We show that the lowest-energy solution of the Hartree-Fock-Bogoliubov (HFB)
equation has the even particle-number parity as long as the time-reversal
symmetry is conserved in the HFB Hamiltonian without null eigenvalues. Based on
this finding, we give a rigorous foundation of a method for solving the HFB
equation to describe the ground state of odd-mass nuclei by employing a
time-reversal anti-symmetric constraint operator to the Hamiltonian, where one
obtains directly the ground state as a self-consistent solution of the
cranked-HFB-type equation. Numerical analysis is done for the neutron-rich Mg
isotopes with a reasonable choice for the operator, and it is demonstrated that
the anomalous increase in the matter radius of $^{37}$Mg is well described when
the last neutron occupies a low angular-momentum orbital in the framework of
the nuclear energy-density-functional method, revealing the deformed halo
structure.
