Interfacial electron-phonon coupling in ultrathin films has attracted much interest; it can give rise to novel effects and phenomena, including enhanced superconductivity. Here we report an observation of strong kinks in the energy dispersions of quantum well states in ultrathin Yb films grown on graphite. These kinks, arising from interfacial electron-phonon coupling, are most prominent for films with a preferred (magic) thickness of 4 monolayers, which are strained and hole doped by the substrate. The energy position of the kinks agrees well with the optical phonon energy of graphite, and the extracted electron-phonon coupling strength {lambda} shows a large subband dependence, with a maximum value up to 0.6. The kinks decay rapidly with increasing film thickness, consistent with its interfacial origin. The variation of {lambda} is correlated with evolution of the electronic wave function amplitudes at the interface. A Lifshitz transition occurs just beyond the magic thickness where the heavy Yb 5d bands begin to populate right below the Fermi level.
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