The clique operator on matching and chessboard graphs

Given positive integers m,nm,n, we consider the graphs GnGn and Gm,nGm,n whose simplicial complexes of complete subgraphs are the well-known matching complex MnMn and chessboard complex Mm,nMm,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,nGm,n in terms of mm and nn, and show that Gm,nGm,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 GnGn.