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This paper addresses the problem of the online communication scheduling and the state estimation for networked control systems with communication constraints, which means that the plant has multiple sensor nodes and only some of them are allowed to communicate with the remote estimator through a limited-bandwidth network at each time instant. By defining a communication logic sequence, a moving-horizon scheduling strategy is first proposed to handle the communication scheduling based on a new quadratic performance criterion including both communication cost and estimation performance penalties, the solving of which can be translated into the optimization problem of the mixed-integer quadratic programming. Subsequently, a moving-horizon state estimator is also addressed, and then, sufficient conditions are presented for the boundness on the square norm of the estimation error. Finally, an experiment on a two-tank liquid-level system is given to demonstrate the validity of the proposed method.