Properties of prime sum graphs

Question

The prime sum graph $P_n$ on the vertex set $V(P_n) = \{1,\dots, n\}$ has an edge $e = xy$ when $x+y$ is prime. It is easy to show that any such $P_n$ is bipartite (put odd numbers in one part and evens in the other). Thus, for a Hamilton cycle to exist, $n$ must be even. So, set $n=2k$. If $2k+1$ and $2k+3$ are prime, then it is easy to see that a Hamilton cycle exists. In the case $n = 10$, then $1,10,3,8,5,6,7,4,9,2,1$ is a Hamilton cycle. Is anything else known about when these graphs are Hamiltonian? What about Eulerian?