We consider the problem of reliably broadcasting information in a multihop asyn- chronous network that is subject to Byzantine failures. That is, some nodes of the network can exhibit arbitrary (and potentially malicious) behavior. Existing solutions provide de- terministic guarantees for broadcasting between all correct nodes, but require that the communication network is highly-connected (typically, 2k + 1 connectivity is required, where k is the total number of Byzantine nodes in the network). In this paper, we investigate the possibility of Byzantine tolerant reliable broadcast be- tween most correct nodes in low-connectivity networks (typically, networks with constant connectivity). In more details, we propose a new broadcast protocol that is specifically designed for low-connectivity networks. We provide sufficient conditions for correct nodes using our protocol to reliably communicate despite Byzantine participants. We present experimental results that show that our approach is especially effective in low-connectivity networks when Byzantine nodes are randomly distributed.