Some classical theorems involving complete graphs have analogues involving complete bipartite graphs. For example, the complete graph $K_n$ has $n^{n-2}$ spanning trees, while the complete bipartite graph $K_{m,n}$ has $n^{m-1} m^{n-1}$ spanning trees. Finding the largest complete subgraph of a graph is a standard NP-hard problem, and finding the largest (in terms of number of edges) complete bipartite subgraph of a bipartite graph is also an NP-hard problem.