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Flooding, that is to deliver a packet from one node to all other nodes in the network, is an integral part of many wireless protocols. Flooding is often implemented by a series of broadcasts of each node and this causes some problems such as the broadcast storm and low reliability, by being engaged with the effects of radio signal propagation, e.g., multipath fading and interference. Many researchers have studied these problems over the years, however, most of these studies have been carried out assuming that all nodes in the network are equipped with a single radio interface and utilize only a single channel. This implies that most of the existing mechanisms that enhance the performance of flooding in wireless networks will not work in multi-radio multi-channel wireless mesh networks (MR-MC WMNs). Motivated by this, in this paper, we propose a flooding mechanism that works well in MR-MC WMNs. Our flooding mechanism, which can operate with an arbitrary number of radio interfaces and channels, increases the reliability of flooding while alleviates the broadcast storm problem, using only local information. Through a detailed simulation study, we demonstrate that our flooding mechanism improves both the reliability and the efficiency of flooding in MR-MC WMNs.