In this paper, we describe the behavior of an edge flame in the mixing layer of two coflowing streams, one of fuel and the other of oxidizer, established at the mouth of a cylindrical injector. The edge flame is stabilized by conductive losses to the rim of the injector and a diffusion flame is trailing behind, either converging to the centerline of the cylinder or extending outward in the oxidizer region. The objective is to determine whether radiative losses alone can drive the edge of the flame to oscillate back and forth, a behavior that has been observed in various experimental configurations. The assumption-of-unity Lewis numbers is adopted to avoid differential-diffusion effects that are known to promote oscillations. Steady and unsteady calculations are reported within a diffusive-thermal approximation, but with finite-rate chemistry. We show that, in the presence of volumetric heat losses, the edge of the flame is stabilized at a larger distance from the rim than in their absence. Furthermore, when the intensity of radiation losses is sufficiently low, the edge remains stationary. When radiative losses become excessive, the edge undergoes sustained oscillations, moving back and forth with a well-defined frequency and amplitude.