  In this work we shall study Einstein Gauss-Bonnet theories and we investigate
when these can have their gravitational wave speed equal to the speed of light,
which is unity in natural units, thus becoming compatible with the striking
event GW170817. We demonstrate how this is possible and we show that if the
scalar coupling to the Gauss-Bonnet invariant is constrained to satisfy a
differential equation, the gravitational wave speed becomes equal to one.
Accordingly, we investigate the inflationary phenomenology of the resulting
restricted Einstein Gauss-Bonnet model, by assuming that the slow-roll
conditions hold true. As we demonstrate, the compatibility with the
observational data coming from the Planck 2018 collaboration, can be achieved,
even for a power-law potential. We restricted ourselves to the study of the
power-law potential, due to the lack of analyticity, however more realistic
potentials can be used, in this case though the calculations are not easy to be
performed analytically. We also pointed out that a string-corrected extension
of the Einstein Gauss-Bonnet model we studied, containing terms of the form
$\sim \xi(\phi) G^{ab}\partial_a\phi \partial_b \phi $ can also provide a
theory with gravity waves speed $c_T^2=1$ in natural units, if the function
$\xi(\phi)$ is appropriately constrained, however in the absence of the
Gauss-Bonnet term $\sim \xi(\phi) \mathcal{G}$ the gravity waves speed can
never be $c_T^2=1$. Finally, we discuss which extensions of the above models
can provide interesting cosmologies, since any combination of $f(R,X,\phi)$
gravities with the above string-corrected Einstein Gauss-Bonnet models can
yield $c_T^2=1$, with $X=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi $.
