This paper studies the event-based bipartite containment control issue for multi-agent networks with cooperation-competition interactions and communication delays. Each agent sends information to the neighbors at a sequence of event-triggered time instants, and updates the controller at its own event times. A novel construction method of virtual unsigned graph and virtual system dynamics is proposed. A specific trigger function to determine the event times is constructed, where both a state-dependent function and an exponentially decay function are jointly introduced as a comparison term. The conditions for convergence of event-triggered algorithms are presented, including the requirement of the graph topology, the system dynamics and parameter selection. A strictly lower bound about the time interval between consecutive events for time $t\in [0,\infty)$ is obtained, and therefore the networked event-triggered system does not exhibit Zeno behavior. In addition, to facilitate the problem that the state dependent trigger function needs continuous information of its neighbors, an alternative trigger function is proposed, which is able to conservatively approximate the evolution of the trigger function using only discrete state information from the neighbors. This avoids continuous communication between agents. Simulations are provided to verify the performance of the proposed event-based bipartite containment control scheme.