Solution:If some NP-complete problem P is in P that P = NP. It simply means that this particular NP-complete problem has a polynomial-time algorithm for a single NP-complete problem, then that would mean all NP problems have polynomial-time solutions, i.e. P would equal NP. This is the definition of NP-completeness.
The Traveling salesman problem (TSP) is indeed in NP. This means that given a proposed solution, verify in polynomial time whether it's a valid solution to the problem. The Boolean satisfiability problem (SAT) is in NP. This means that given a proposed solution to a Boolean formula, verify in polynomial time whether is satisfies the formula. However, Similar to TSP whether SAT is in P.
Hamiltonian cycle if graph G contains a Hamiltonian path. Therefore, any instance of the Hamiltonian cycle problem can be reduced to an instance of the Hamiltonian path problem. Thus, the Hamiltonian cycle is NP- Hard. The Hamiltonian cycle is both, a NP. problem and NP-Hard. Therefore, it is a NP-complete problem.
So, the statement A, B and C are correct and statement D is incorrect.