We propose novel centralized minimum span frequency assignment (MS-FA) techniques allowing partially overlapped channels (POCs) to improve cost and spectral efficiency for integrated access and backhaul (IAB) networks. We first revisit the conventional MS-FA problem (MS-FAP) that minimizes the required bandwidth for given quality-of-service (QoS) requirements, and then design a novel integer linear programming (ILP)-based optimal MS-FA technique with POCs. However, the optimal technique may not be feasible for large-scale IAB networks due to its NP-hardness. Hence, we propose a novel low-complexity MS-FA algorithm that achieves a near-optimal performance while significantly reducing the computational complexity. To the best of our knowledge, no MS-FA technique exists to consider both POCs and the physical interference model in the literature. Simulation results show that the spectral efficiency becomes significantly improved by reducing the required bandwidth in practical IAB networks.