This study focuses on the thermal-diffusive oscillations of the full spectrum of co-flow laminar flames-from premixed to non-premixed through partially premixed flames for Lewis numbers sufficiently greater than unity. A premixedness parameter is identified using the analytical solution to the mixing field downstream of a splitter plate. Different premixedness levels ranging from zero (fully non-premixed) to unity (fully premixed) are then used as Dirichlet data for inlet concentrations of the reactants to the flame. The initial value problem is numerically solved by adopting a thermo-diffusive model and single-step finite rate chemical reaction and using higher-order compact schemes. A thumb-shaped region in the Damköhler number - premixedness parameter space is identified as corresponding to limit cycle oscillations of the flame. This fills the gap between what is reported for the pulsating instability of non-premixed edge flames recently and fully premixed flames earlier. The thumb-shaped region enlarges with increase in the Lewis number in the range of 1.5-1.7, with marked increase in the oscillatory amplitude of the total heat release. A subcritical Hopf bifurcation is seen to occur at the boundary of this region in the above parametric space. The mechanism of oscillations is studied: hysteresis is found between the upstream and downstream propagation of the flame due to the thermal-diffusive imbalance. The oscillations are plotted in the phase space of the domain-integrated mass and heat diffusive fluxes of and to the reactants. The area of the limit cycle peaks at a particular Damköhler number and premixedness parameter and drops to zero on both sides in this parametric space. © 2012 The Combustion Institute.