In this article, we study communication-constrained networked control problems for linear time-invariant systems in the presence of Denial-of-Service (DoS) attacks, namely attacks that prevent transmissions over the communication network. Our article aims at exploring the tradeoffs between system resilience and network bandwidth capacity. Given a class of DoS attacks, we characterize the bit-rate conditions that are dependent on the unstable eigenvalues of the dynamic matrix of the plant and the parameters of DoS attacks, under which exponential stability of the closed-loop system can be guaranteed. Our characterization clearly shows the tradeoffs between the communication bandwidth and resilience against DoS. An example is given to illustrate the proposed approach.