Abstract: The directed Hamilton cycle problem is investigated based on automata theory and a polynomial algorithm is proposed to test the sufficient condition of this problem. More specifically, the digraph is modeled as an automaton, and the concept of Hamiltonian graph is formalized based on automaton. Then a polynomial algorithm is proposed to check whether a subset of a language marked by an automaton satisfies a sufficient condition of true subset. Based on this algorithm, a polynomial algorithm is proposed to test whether a digraph satisfies a sufficient condition of Hamiltonian graph and find the corresponding Hamilton cycles. In particular, a sufficient condition for checking whether a digraph is a Hamiltonian digraph and a sufficient condition for checking whether a cycle in a digraph is a Hamilton cycle are given.