Hamiltonicity in Prime Sum Graphs

Abstract

For any positive integer n , we define the prime sum graph $$G_n=(V,E)$$ G n = ( V , E ) of order n with the vertex set $$V=\\{1,2,\\cdots , n\\}$$ V = { 1 , 2 , ⋯ , n } and $$E=\\{ij: i+j \\text{ is } \\text{ prime }\\}$$ E = { i j : i + j is prime } . Filz in 1982 posed a conjecture that $$G_{2n}$$ G 2 n is Hamiltonian for any $$n\\ge 2$$ n ≥ 2 , i.e., the set of integers $$\\{1,2,\\cdots , 2n\\}$$ { 1 , 2 , ⋯ , 2 n } can be represented as a cyclic rearrangement so that the sum of any two adjacent integers is a prime number. With a fundamental result in graph theory and a recent breakthrough on the twin prime conjecture, we prove that Filz’s conjecture is true for infinitely many cases.