  Background. The Gorkov approach to self-consistent Green's function theory
has been formulated in [V. Som\`a, T. Duguet, C. Barbieri, Phys. Rev. C 84,
064317 (2011)]. Over the past decade, it has become a method of reference for
first-principle computations of semi-magic nuclear isotopes. The currently
available implementation is limited to a second-order self-energy and neglects
particle-number non-conserving terms arising from contracting three-particle
forces with anomalous propagators. For nuclear physics applications, this is
sufficient to address first-order energy differences, ground-state radii and
moments on an accurate enough basis. However, addressing absolute binding
energies, fine spectroscopic details of $N\pm1$ particle systems or delicate
quantities such as second-order energy differences associated to pairing gaps,
requires to go to higher truncation orders.
  Purpose. The formalism is extended to third order in the algebraic
diagrammatic construction (ADC) expansion with two-body Hamiltonians.
  Methods. The expansion of Gorkov propagators in Feynman diagrams is combined
with the algebraic diagrammatic construction up to the third order as an
organization scheme to generate the Gorkov self-energy.
  Results. Algebraic expressions for the static and dynamic contributions to
the self-energy, along with equations for the matrix elements of the Gorkov
eigenvalue problem, are derived. It is first done for a general basis before
specifying the set of equations to the case of spherical systems displaying
rotational symmetry. Workable approximations to the full self-consistency
problem are also elaborated on. The formalism at third order it thus complete
for a general two-body Hamiltonian.
  Conclusion. Working equations for the full Gorkov-ADC(3) are now available
for numerical implementation.
