The multi-robot coverage motion planning (MCMP) problem in which every reachable area must be covered is common in multi-robot systems. To deal with the MCMP problem, we propose an efficient, complete, and off-line algorithm, named the “auction-based spanning tree coverage (A-STC)” algorithm. First, the configuration space is divided into mega cells whose size is twice the minimum coverage range of a robot. Based on connection relationships among mega cells, a graph structure can be obtained. A robot that circumnavigates a spanning tree of the graph can generate a coverage trajectory. Then, the proposed algorithm adopts an auction mechanism to construct one spanning tree for each robot. In this mechanism, an auctioneer robot chooses a suitable vertex of the graph as an auction item from neighboring vertexes of its spanning tree by heuristic rules. A bidder robot submits a proper bid to the auctioneer according to the auction vertexes’ relationships with the spanning tree of the robot and the estimated length of its trajectory. The estimated length is calculated based on vertexes and edges in the spanning tree. The bidder with the highest bid is selected as a winner to reduce the makespan of the coverage task. After auction processes, acceptable coverage trajectories can be planned rapidly. Computational experiments validate the effectiveness of the proposed MCMP algorithm and the method for estimating trajectory lengths. The proposed algorithm is also compared with the state-of-the-art algorithms. The comparative results show that the A-STC algorithm has apparent advantages in terms of the running time and the makespan for large crowded configuration spaces.