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 cliqueHelly. If the parameters are small enough, we show that these graphs (even if not cliqueHelly) 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 cliqueHelly. We give partial results for the clique behavior of the matching graph Gn.