Hamiltonian Cycle Problem is in P

In this paper we present the first deterministic polynomial time algorithm for determining the existence of a Hamiltonian cycle and finding a Hamiltonian cycle in general graphs. Our algorithm can also solve the Hamiltonian path problem in the traceable graphs. The space complexity of our algorithm is O(n^4). The time complexity are theoretically O(n^5*d2) on average and O(n^6*d2) in the worst case respectively, where d is the maximum degree of vertex. With parallel computing, the space complexity can be improved to O(n³⁾ and the time complexity to O(n^3*d2) on average and O(n^4*d2) in the worst case. We construct the corresponding path hologram transformed from the original graph and compute the path set, which is a collection of segment sets consisting of all the vertices located on the same segment level among all the longest basic paths, of every vertex with greedy strategy. The path hologram is a multi-segment graph with the vertex <u, k> where u is a vertex and k is the segment level of u in the path hologram. To ensure that each valid path fragments can be visited and invalid path fragments cannot be visited, the key strategy of our method is the "consecutive" deleting-replenishing operations recursively on the left/right action field of a vertex, respectively. In fact, our algorithm can be directly applied to the original graph. Besides, our algorithm can deal with the finite general graphs including undirected, directed, and mixed. As a result, the well-known problem HCP in NPC can be now solved practically in deterministic polynomial time for general graphs in the worst case.