This paper considers the Byzantine consensus problem for nodes with binary inputs. The nodes are interconnected by a network represented as an undirected graph, and the system is assumed to be synchronous. Under the classical point-to-point communication model, it is well-known [7] that the following two conditions are both necessary and sufficient to achieve Byzantine consensus among $n$ nodes in the presence of up to $f$ Byzantine faulty nodes: $n \ge 3f+1$ and vertex connectivity at least $2f+1$. In the classical point-to-point communication model, it is possible for a faulty node to equivocate, i.e., transmit conflicting information to different neighbors. Such equivocation is possible because messages sent by a node to one of its neighbors are not overheard by other neighbors. This paper considers the local broadcast model. In contrast to the point-to-point communication model, in the local broadcast model, messages sent by a node are received identically by all of its neighbors. Thus, under the local broadcast model, attempts by a node to send conflicting information can be detected by its neighbors. Under this model, we show that the following two conditions are both necessary and sufficient for Byzantine consensus: vertex connectivity at least $\lfloor 3f/2 \rfloor + 1$ and minimum node degree at least $2f$. Observe that the local broadcast model results in a lower requirement for connectivity and the number of nodes $n$, as compared to the point-to-point communication model. We extend the above results to a hybrid model that allows some of the Byzantine faulty nodes to equivocate. The hybrid model bridges the gap between the point-to-point and local broadcast models, and helps to precisely characterize the trade-off between equivocation and network requirements.