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The striking GW170817 event indicated that the graviton is nearly massless,since the gamma rays emitted from the two neutron stars merging arrived almostsimultaneously with the gravitational waves. Thus, the graviton must also bemassless during the inflationary and post-inflationary era, since there is noobvious reason to believe otherwise. In this letter we shall investigate thetheoretical implications of the constraint that the graviton is massless to anEinstein-Gauss-Bonnet theory with linear coupling of the scalar field to thefour dimensional Gauss-Bonnet invariant. As we show, the constraint of havinggravitational wave speed of the primordial gravitational waves equal to one,severely restricts the dynamics of the scalar field, imposing a directconstant-roll evolution on it. Also, as we show, the spectral index of theprimordial scalar perturbations for the GW170817-compatibleEinstein-Gauss-Bonnet theory with linear coupling is different in comparison tothe same theory with non-linear coupling. Thus the phenomenology of the modelis expected to be different, and we briefly discuss this issue too. Inaddition, the constant-roll condition is always related to non-Gaussianities,thus it is interesting that the imposition of a massless graviton in anEinstein-Gauss-Bonnet theory with linear coupling may lead tonon-Gaussianities, so we briefly discuss this issue too.

A Nearly Massless Graviton in Einstein-Gauss-Bonnet Inflation with Linear Coupling Implies Constant-roll for the Scalar Field