  In this work we study the inflationary framework of Einstein-Gauss-Bonnet
theories which produce a primordial gravitational wave speed that respects the
constraint imposed by the GW170817 event, namely $\left| c_T^2 - 1 \right| < 6
\times 10^{-15}$ in natural units. For these theories in general, the scalar
Gauss-Bonnet coupling function $\xi (\phi)$ is arbitrary and unrelated to the
scalar potential. We develop the inflationary formalism of this theory, and
present analytic forms for the slow-roll indices, the observational indices and
the $e$-foldings number, assuming solely a slow-roll era and also that the
constraints imposed by the GW170817 event hold true. We present in detail an
interesting class of models that produces a viable inflationary era. Finally,
we investigate the behavior of the Einstein-Gauss-Bonnet coupling function
during the reheating era, in which case the evolution of the Hubble rate
satisfies a constant-roll-like relation $\dot{H}=\delta H^2$. As we show, the
behavior of the scalar Gauss-Bonnet coupling during the reheating is different
from that of the inflationary era. This is due to the fact that the evolution
governing the reheating is different compared to the inflationary era.
