  A consequence of Bertrand's postulate, proved by L. Greenfield and S.
Greenfield in 1998, assures that the set of integers $\{1,2,\cdots, 2n\}$ can
be partitioned into pairs so that the sum of each pair is a prime number for
any positive integer $n$. Cutting through it from the angle of Graph Theory,
this paper provides new insights into the problem. We conjecture a stronger
statement that the set of integers $\{1,2,\cdots, 2n\}$ can be rearranged into
a cycle so that the sum of any two adjacent integers is a prime number. Our
main result is that this conjecture is true for infinitely many cases.
