The clique operator on matching and chessboard graphs

Given positive integers m, n, we consider the graphs Gn and Gm,n whose simplicial complexes of complete subgraphs are the well-known matching complex Mn and chessboard complex Mm,n. Those are the matching and chessboard graphs. We determine which matching and chessboard graphs are clique-Helly. If the parameters are small enough, we show that these graphs (even if not clique-Helly) are homotopy equivalent to their clique graphs. We determine the clique behavior of the chessboard graph Gm,n in terms of m and n, and show that Gm,n is clique-divergent if and only if it is not clique-Helly. We give partial results for the clique behavior of the matching graph Gn.