Hamiltonicity in Prime Sum Graphs

For any positive integer n , we define the prime sum graph G n = ( V , E ) of order n with the vertex set V = { 1 , 2 , ⋯ , n } and E = { i j : i + j is prime } . Filz in 1982 posed a conjecture that G 2 n is Hamiltonian for any n ≥ 2 , i.e., the set of integers { 1 , 2 , ⋯ , 2 n } can be represente...

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Bibliographic Details
Published inGraphs and combinatorics Vol. 37; no. 1; pp. 209 - 219
Main Authors Chen, Hong-Bin, Fu, Hung-Lin, Guo, Jun-Yi
Format Journal Article
LanguageEnglish
Published Tokyo Springer Japan 2021
Springer Nature B.V
Subjects
Combinatorics
Engineering Design
Graph theory
Integers
Mathematics
Mathematics and Statistics
Original Paper
Prime numbers
Vertex sets
Prime sum graph
Hamilton cycle
05C45
Filz’s conjecture
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Summary:For any positive integer n , we define the prime sum graph G n = ( V , E ) of order n with the vertex set V = { 1 , 2 , ⋯ , n } and E = { i j : i + j is prime } . Filz in 1982 posed a conjecture that G 2 n is Hamiltonian for any n ≥ 2 , i.e., the set of integers { 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.
ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-020-02241-1