Cluster states are an essential component in one-way quantum computation protocols. We present two schemes to generate addressable continuous-variable cluster states from quadrature squeezed cylindrically polarized modes. By including polarization in
addition to the transverse spatial degree of freedom, elementary cluster states can be created in which four cluster nodes co-propagate within one paraxial vector beam. This approach is fundamentally compatible with existing time-multiplexed schemes that have been used to create the largest cluster states to date. We implement a proof-of-principle experiment of one of the proposed schemes and verify its feasibility by measuring the quantum correlations between the different nodes of the cluster state.
