Multi-radio multi-channel (MRMC) wireless mesh networks (WMNs) achieve higher throughput using multiple simultaneous transmissions and receptions. However, due to limited number of non-overlapping channels, such networks suffer from co-channel interference, which degrades their performance. To mitigate co-channel interference, effective channel assignment algorithms (CAAs) are desired. In this article, we propose a novel CAA, Topology-controlled Interference-aware Channel-assignment Algorithm (TICA), for MRMC WMNs. This algorithm uses topology control based on power control to assign channels to multi-radio mesh routers such that co-channel interference is minimized, network throughput is maximized, and network connectivity is guaranteed. We further propose to use two-way interference-range edge coloring, and call the improved algorithm Enhanced TICA (e-TICA), which improves the fairness among flows in the network. However, the presence of relatively long links in some topologies leads to conflicting channel assignments due to their high interference range. To address this issue, we propose to utilize minimum spanning tree rooted at the gateway to reduce conflicting channels, and in turn, improve medium access fairness among the mesh nodes. We call the improved algorithm e-TICA version 2 (e-TICA2). We evaluate the performance of the proposed CAAs using simulations in NS2. We show that TICA significantly outperforms the Common Channel Assignment scheme in terms of network throughput, and e-TICA and e-TICA2 achieve better fairness among traffic flows as compared to TICA. It is also shown that e-TICA2 leads to improved network throughput, as compared to TICA and e-TICA.