Polycyclic aromatic hydrocarbon (PAH) growth is known to be the principal pathway to soot formation during fuel combustion, as such, a physical understanding of the PAH growth mechanism is needed to effectively assess, predict, and control soot formation in flames. Although the hydrogen abstraction C2H2 addition (HACA) mechanism is believed to be the main contributor to PAH growth, it has been shown to under-predict some of the experimental data on PAHs and soot concentrations in flames. This article presents a submechanism of PAH growth that is initiated by propargyl (C3H3) addition onto naphthalene (A2) and the naphthyl radical. C3H3 has been chosen since it is known to be a precursor of benzene in combustion and has appreciable concentrations in flames. This mechanism has been developed up to the formation of pyrene (A4), and the temperature-dependent kinetics of each elementary reaction has been determined using density functional theory (DFT) computations at the B3LYP/6-311++G(d,p) level of theory and transition state theory (TST). H-abstraction, H-addition, H-migration, β-scission, and intramolecular addition reactions have been taken into account. The energy barriers of the two main pathways (H-abstraction and H-addition) were found to be relatively small if not negative, whereas the energy barriers of the other pathways were in the range of (6-89 kcal·mol(-1)). The rates reported in this study may be extrapolated to larger PAH molecules that have a zigzag site similar to that in naphthalene, and the mechanism presented herein may be used as a complement to the HACA mechanism to improve prediction of PAH and soot formation.