This paper investigates trajectories within the Alpha Centauri system to reach planet Proxima b. These trajectories come in the form of connections between the classical Lagrange points of Alpha-Centauri’s binary system (composed of the stars Alpha Centauri A and B, AC-A and AC-B) and the classical Lagrange points of the Alpha Centauri C (AC-C)/Proxima b system. These so-called heteroclinic connections are sought using a patched restricted three-body problem method. A genetic algorithm is applied to optimize the linkage conditions between the two three-body systems, focusing on minimizing the position, velocity, and time error at linkage. Four different futuristic, graphenebased sail configurations are used for the analyses: two sails with a reflective coating on only one side of the sail with lightness numbers equal to β = 100 and β = 1779, and two sails with a reflective coating on both sides (again, considering β = 100 and β = 1779). Results from the genetic algorithm show that, for example, a transfer from the L2-point in the AC-A/AC-B system to the L1-point in the AC-C/Proxima b system can be accomplished with a transfer time of 235 years for the one-sided graphene-based sail with β = 1779.