A complete bipartite graph, sometimes also called a complete bicolored graph (Erdős et al. 1965) or complete bigraph, is a bipartite graph (i.e., a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent) such that every pair of graph vertices in the two sets are adjacent. If there are and graph vertices in the two sets, the complete bipartite graph is denoted . The above figures show and . is also known as the utility graph (and is the circulant graph ), and is the unique 4-cage graph. is a Cayley graph. A complete bipartite graph is a circulant graph (Skiena 1990, p. 99), specifically , where is the floor function.Special cases of are summarized in the table below.path graph path graph claw graphstar graph square graph utility graphThe numbers of (directed) Hamiltonian cycles for the graph with , 2, ... are 0, 2, 12, 144, 2880, 86400, 3628800, 203212800, ... (OEIS A143248), where..