Simultaneous localization and mapping (SLAM) technology provides basic location services and environment sensing for underground coal mine rescue. Due to the low illumination and less texture underground environment, as well as the restricted computing resources, existing SLAM systems are prevented from working stably in real-time on mobile devices. In this paper, we propose a fast optical flow matching SLAM (FOFM-SLAM) based on edge computing in underground rescue environments. This is the first edge-assisted SLAM system that adopts a two-stage sparse optical flow tracking method based on image pyramid for feature matching and refines the keypoint correspondences by outlier filtering strategy. Further, we design a keyframe selection strategy based on limited viewpoint transfer, taking into account factors such as parallax and tracking points to ensure tracking stability. We perform comprehensive experiments on TUM and ETH3D RGB-D datasets and fully implement FOFM-SLAM on various types of devices. Results reveal that FOFM-SLAM achieves a relative pose error of 0.51cm, and the tracking time is improved by about 40% on the ETH3D dataset compared with the existing solutions. Finally, we implement the field tests in the coal mine to present the practicality.