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Nowadays wireless mesh routers are facilitating with more wireless channels than ever because of the advanced wireless communication technologies such as OFDM, SDR and CR(cognitive radio). With multi-radio and multichannel communication capability, spectrum heterogeneity are widely existing in multi-radio wireless systems, such that network performances usually suffer from channel bandwidth and transmission power discrepancy. Many seminar works having been proposed are generally focusing on throughput maximization in a given network configuration, which would suffer from highly dynamic network topology and variable spectrum conditions. In this paper, we prove that, heterogeneity in wireless network would lead to resource constrained regions, and unfair wireless resource allocation would deteriorate throughput on some links. In this paper, we propose a fair and stable scheduling algorithm on heterogeneous multichannel multi-radio wireless mesh network (MCMR WMN), balancing these two objectives between network throughput. Our scheduling algorithm is based on multiple coloring algorithm and maximum matching algorithm, where wireless channels and interference links correspond to the two node sets in bipartite graph. Multiple channels and interfaces would be effectively colored in achieving a minimal scheduling period. Analytical results show that, max-min fairness could optimize heterogeneous network throughput in case that the data flow is congested. Stable scheduling would degrade the throughput and fairness as well, however, it could maintain a relative high throughput as channels would possibly fail. Our algorithm is effective in achieving optimized network throughput while maintaining a relative stability. Simulation results show that , the proposed scheduling algorithm effectively balance between fairness and stability on wireless channel utilization.