Abstract. The Hamiltonian cycle problem (HCP) in digraphs D with degree bound two is solved by two mappings in this paper. The first bijection is between an incidence matrix Cnm of simple digraph and an incidence matrix F of balanced bipartite undirected graph G; The second mapping is from a perfect matching of G to a cycle of D. It proves that the complexity of HCP in D is polynomial, and finding a second non-isomorphism Hamiltonian cycle from a given Hamiltonian digraph with degree bound two is also polynomial. Lastly it deduces P = NP base on the results. 