Pls help! Full solutions thank you!

The Hamiltonian Path Problem is a famous algorithmic problem on graphs. Given a graph G with n vertices, the objective is to find a path containing every vertex exactly once (or decide that no such path exists). In other words, the objective is to find a sequence a1,...an of integers such that each integer in {1, ... , n} appears exactly once in the sequence, there is an edge between vertex a, and vertex ai+1 for all isi<.>