This paper presents results of a Monte Carlo simulation for the glass transition in a three-dimensional polymer melt. The melt was simulated by the bond-fluctuation model on a simple cubic lattice, which was amended by a two-level Hamiltonian favoring long bonds in order to generate a competition between the energetic constraints and the density of the melt. The development of this competition during the cooling process makes the melt adopt configurations, from which it cannot easily relax, and thus facilitates the freezing of the melt in an amorphous structure, as soon as the internal relaxation times match the observation time of the simulation set by the cooling rate. How pronounced the effects of this competition are depends upon the chain length, whose influence on the vitrification process we want to study by monitoring various quantities that probe different length scales of a polymer, such as the mean bond length or the radius of gyration. As the melt vitrifies, these quantities gradually become independent of temperature, and their value at low temperatures is strongly influenced by the chain length. This influence qualitatively resembles that of a variation of the cooling rate. The larger the chain length (cooling rate), the faster the various quantities fall out of equilibrium, and the earlier the melt freezes on the corresponding length scale. It is thus possible to infer from these quantities the chain-length dependence of the glass transition temperature T(g). This analysis shows that T(g) approximately increases linearly with the inverse chain length.