  In this paper, we study the Gaussian relay channels in the low
signal-to-noise ratio (SNR) regime with the time-sharing compress-and-forward
(CF) scheme, where at each time slot all the nodes keep silent at the first
fraction of time and then transmit with CF at a higher peak power in the second
fraction. Such a silent vs. active two-phase relay scheme is preferable in the
low-SNR regime. With this setup, the upper and lower bounds on the minimum
energy per bit required over the relay channel are established under both
full-duplex and half-duplex relaying modes. In particular, the lower bound is
derived by applying the max-flow min-cut capacity theorem; the upper bound is
established with the aforementioned time-sharing CF scheme, and is further
minimized by letting the active phase fraction decrease to zero at the same
rate as the SNR value. Numerical results are presented to validate the
theoretical results.
