MazeGraph[m_, n_] :=
 Block[{$RecursionLimit = Infinity, grid = GridGraph[{m, n}],
   unvisitedQ}, unvisitedQ[_] := True;
  Graph[Range[m n], Reap[{unvisitedQ[#] = False;
        Do[
         If[unvisitedQ[neighbor],
          Sow[# <-> neighbor]; #0@neighbor], {neighbor,
          RandomSample@AdjacencyList[grid, #]}]} &@
      RandomChoice@VertexList@grid][[2, 1]],
   GraphLayout -> {"GridEmbedding", "Dimension" -> {m, n}}]];
maze = MazeGraph[13, 21]