We study a combustion-radiation model which models premixed flames propagating in a gaseous mixture with inert dust. This model combines diffusion of mass and temperature with reaction at the flame front. We choose a free boundary model to describe the propagating flames and take a linearized approximation to model the radiation, but we keep a nonlinear reaction term which is temperature dependent. The radiative transfer of thermal energy emitted and absorbed by dust is modelled using the Eddington equation.We analyse the bifurcation diagram of the travelling wave solution curve. In a specific parameter plane, travelling waves are given by a single smooth curve which is parameterized by the flame temperature. © 2011 Taylor & Francis.