Angle-resolved photoemission spectroscopy reveals pronounced kinks in the dispersion of the σ band of graphene. Such kinks are usually caused by the combination of a strong electron-boson interaction and the cutoff in the Fermi-Dirac distribution. They are therefore not expected for the σ band of graphene that has a binding energy of more than ≈3.5 eV. We argue that the observed kinks are indeed caused by the electron-phonon interaction, but the role of the Fermi-Dirac distribution cutoff is assumed by a cutoff in the density of σ states. The existence of the effect suggests a very weak coupling of holes in the σ band not only to the π electrons of graphene but also to the substrate electronic states. This is confirmed by the presence of such kinks for graphene on several different substrates that all show a strong coupling constant of λ≈1.