We analyze the critical transitions (a) to endemic states in an SIS epidemiological model, and (b) to full synchronization in an ensemble of coupled chaotic maps, on networks where, at any given time, each node is connected to just one neighbour. In these “monogamous” populations, the lack of connectivity in the instantaneous interaction pattern—that would prevent both the propagation of an infection and the collective entrainment into synchronization—is compensated by occasional random reconnections which recombine interacting couples by exchanging their partners. The transitions to endemic states and to synchronization are recovered if the recombination rate is sufficiently large, thus giving rise to a bifurcation as this rate varies. We study this new critical phenomenon both analytically and numerically.