We propose Low Density Parity Check (LDPC) code designs for the half-duplex relay channel. Our designs are based on the information theoretic random coding scheme for decode-and-forward relaying. The source transmission is decoded with the help of side information in the form of additional parity bits from the relay. We derive the exact relationships that the component LDPC code profiles in the relay coding scheme must satisfy. These relationships act as constraints for the density evolution algorithm which is used to search for good relay code profiles. To speed up optimization, we outline a Gaussian approximation of density evolution for the relay channel. The asymptotic noise thresholds of the discovered relay code profiles are a fraction of a decibel away from the achievable lower bound for decode-and-forward relaying. With random component LDPC codes, the overall relay coding scheme performs within 1.2 dB of the theoretical limit.