n-Rooks and n-queens problem on planar and modular chessboards with hexagonal cells

We show the existence of solutions to the n-rooks problem and n-queens problem on chessboards with hexagonal cells, problems equivalent to certain three and six direction riders on ordinary chessboards. Translating the problems into graph theory problems, we determine the independence number (maximum size of independent set) of rooks graph and queens graph. We consider the $n \times n$ planar diamond-shaped H_n with hexagonal cells, and the board $H_n$ as a flat torus $T_n$. Here, a rook can execute moves on lines perpendicular to the six sides of the cell it is placed, and a queen can execute moves on those lines together with lines through the six corners of the cell it is placed.