given coordinates , where and . Two vertices with coordinates and are adjacent if and only if either (the two squares belong to the same file of the chessboard, and are connected by a vertical rook move) or (the two squares belong to the same rank and are connected by a horizontal move). Within a single rank or a single file of the chessboard, all squares are reachable from each other, so these squares form a clique, a subset of vertices forming a complete graph. The whole rook's graph for an chessboard can be formed from these two kinds of cliques as the Cartesian product of graphs . Because the rook's graph for a square chessboard is the Cartesian product of equal-size cliques, it is an example of a Hamming graph. Its dimension as a Hamming graph is two, and every two-dimensional Hamming graph is a rook's graph for a square chessboard. Square rook's graphs are also called Latin square graphs, because the vertices of such a graph describe the squares of a Latin square and its edges describe pairs of squares that cannot contain the same value. The Sudoku graphs are supergraphs of rook's graphs with some additional edges, connecting squares of a Sudoku puzzle that should have unequal values. Geometrically, the rook's graphs can be formed by sets of the vertices and edges (the skeletons) of a family of convex polytopes, the Cartesian products of pairs of neighborly polytopes. For instance, the 3-3 duoprism is a four-dimensional shape formed as the Cartesian product of two triangles, and has a rook's graph as its skeleton. Regularity and symmetry Strong regularity and observe that the rook's graph (or equivalently, as they describe it, the line graph of the complete bipartite graph ) has all of the following properties: It has vertices, one for each square of the chessboard. Each vertex is adjacent to edges, connecting it to the squares on the same rank and the squares on the same file. The triangles within the rook's graph are formed by triples of squares within a single rank or file. When , exactly edges (the ones connecting squares on the same rank) belong to triangles; the remaining edges (the ones connecting squares on the same file) belong to triangles. When , each edge belongs to triangles. Every two vertices that are not adjacent to each other belong to a unique -vertex cycle,