Consider the following problem: Given a weighted directed graph $G$, determine if $G$ has a cycle whose total weight is $k$. All edge weights are integer but might be negative. $k$ is not an inputted value. It's fixed and known beforehand to everyone. So $k$ is NOT an input to this algorithm.
I want to show the problem is NP hard, so this means I have an oracle that tells me whether or not a graph has a cycle whose sum of edge weights equals $k$ for a fixed, known value of $k$.
I thought about reducing from Hamiltonian cycle. So I want to show that I can use this oracle to solve Hamiltonian cycle problems.
Let $G$ be a graph. I want to see if $G$ has a Hamiltonian cycle. I try to construct a new graph to provide to the oracle but I'm not sure how to do so. I think it needs to have $k$ vertices so I tried doing casework on when $G$ has more or less than this many vertices.
I can show that I can solve Hamiltonian cycle problems with $k$ vertices by constructing a new complete graph where edge weights are $1$ if it was in the original graph and a really large nunber otherwise. Then running the oracle on this graph returns true only if there's a Hamiltonian cycle with $k$ vertices. But this doesn't handle the general case. Is this proof fine?
But I'm stuck. Any help is appreciated.