 They may be characterized in terms of the number of triangles each edge belongs to and by the existence of a -cycle connecting each nonadjacent pair of vertices.Rook's graphs are perfect graphs, meaning that their induced subgraphs (the line graphs of bipartite graphs) all have chromatic number equal to their clique number.The induced subgraphs of rook's graphs form one of the key components of a decomposition of perfect graphs used to prove the strong perfect graph theorem characterizing all perfect graphs.