We consider resilient versions of discrete-time multi-agent consensus in the presence of faulty or even malicious agents in the network. In particular, we develop event-triggered update rules which can mitigate the influence of the malicious agents and at the same time reduce the necessary communication. Each regular agent updates its state based on a given rule using its neighbors' information. Only when the triggering condition is satisfied, the regular agents send their current states to their neighbors. Otherwise, the neighbors will continue to use the state received the last time. Assuming that a bound on the number of malicious nodes is known, we propose two update rules with event-triggered communication. They follow the so-called mean subsequence reduced (MSR) type algorithms and ignore values received from potentially malicious neighbors. We provide full characterizations for the necessary connectivity in the network for the algorithms to perform correctly, which are stated in terms of the notion of graph robustness. A numerical example is provided to demonstrate the effectiveness of the proposed approach.