Mathematical chess problems | Parametric families of graphs
Queen's graph
In mathematics, a queen's graph is a graph that represents all legal moves of the queen—a chess piece—on a chessboard. In the graph, each vertex represents a square on a chessboard, and each edge is a legal move the queen can make, that is, a horizontal, vertical or diagonal move by any number of squares. If the chessboard has dimensions , then the induced graph is called the queen's graph. Independent sets of the graphs correspond to placements of multiple queens where no two queens are attacking each other. They are studied in the eight queens puzzle, where eight non-attacking queens are placed on a standard chessboard. Dominating sets represent arrangements of queens where every square is attacked or occupied by a queen; five queens, but no fewer, can dominate the chessboard. Colourings of the graphs represent ways to colour each square so that a queen cannot move between any two squares of the same colour; at least n colours are needed for an chessboard, but 9 colours are needed for the board. (Wikipedia).
