I have an infinite sequence of simple connected undirected self-adjacency-free graphs $g_{n}$ for $n\ge3$, with the properties that $g_n$ has exactly $n$ vertices and also contains $g_{n-1}$ as a subgraph i.e. has one extra node, $V$, compared to $g_{n-1}$. The degree of $V$ is non-strictly increasing with $n$, but which existing vertices those new edges connect to is unpredictable - more specifically, an edge exists between vertices $i,j$ when $i+j$ is prime.
Is there anything I can say about whether these graphs contain Hamiltonian paths, given earlier ones do? Ideally, I would like to somehow prove that if a path exists for some value of $n$, this implies that a path also exists for all greater values, but I'm not sure how to achieve that.
I can see from how the degree of $V$ works that if I can somehow prove that $V$ having "enough" edges guarentees that an existing path can be extended to $n$ nodes from $n-1$, that solves my problem. I can also see that if I can prove $V$ is a neighbour to two nodes that are neighbours of each other, I can extend the path that way, but don't know how to prove that either.