European Journal of Mathematics

, Volume 3, Issue 2, pp 379–386 | Cite as

Complete subgraphs of the coprime hypergraph of integers I: introduction and bounds

  • Jan-Hendrik de WiljesEmail author
Research Article


We introduce the uniform coprime hypergraph of integers \(\mathrm{CHI}_k\) which is the graph with vertex set \({\mathbb {Z}}\) and a Open image in new window -hyperedge exactly between every \(k+1\) elements of \({\mathbb {Z}}\) having greatest common divisor equal to 1. This generalizes the concept of coprime graphs. We obtain some basic properties of these graphs and give upper and lower bounds for the clique number of certain subgraphs of \(\mathrm{CHI}_k\).


Hypergraphs on integers Clique number Degree sequence 

Mathematics Subject Classification

11B75 05C65 05C69 


  1. 1.
    Ahlswede, R., Khachatrian, L.H.: On extremal sets without coprimes. Acta Arith. 66(1), 89–99 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ahlswede, R., Khachatrian, L.H.: Maximal sets of numbers not containing \(k + 1\) pairwise coprime integers. Acta Arith. 72(1), 77–100 (1995)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ahlswede, R., Khachatrian, L.H.: Sets of integers and quasi-integers with pairwise common divisor. Acta Arith. 74(2), 141–153 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ahlswede, R., Khachatrian, L.H.: Sets of integers with pairwise common divisor and a factor from a specified set of primes. Acta Arith. 75(3), 259–276 (1996)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics, vol. 244. Springer, New York (2008)zbMATHGoogle Scholar
  6. 6.
    Dusart, P.: The \(k{\rm th}\) prime is greater than \(k(\ln k+\ln \ln k-1)\) for \(k\geqslant 2\). Math. Comp. 68(225), 411–415 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Erdős, P.: Remarks in number theory. IV. Extremal problems in number theory. I. Mat. Lapok 13, 228–255 (1962) (in Hungarian)Google Scholar
  8. 8.
    Erdős, P., Gallai, T.: Graphs with prescribed degrees of vertices. Mat. Lapok 11, 264–274 (1960) (in Hungarian)Google Scholar
  9. 9.
    Erdős, P., Sarkozy, G.N.: On cycles in the coprime graph of integers. Electron. J. Combin. 4(2), #R8 (1997)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Haxell, P., Pikhurko, O., Taraz, A.: Primality of trees. J. Comb. 2(4), 481–500 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Mutharasu, S., Mohamed Rilwan, N., Angel Jebitha, M.K., Tamizh Chelvam, T.: On generalized coprime graphs. Iran. J. Math. Sci. Inform. 9(2), 1–6 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Pomerance, C., Selfridge, J.L.: Proof of D.J. Newman’s coprime mapping conjecture. Mathematika 27(1), 69–83 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Robertson, L., Small, B.: On Newman’s conjecture and prime trees. Integers 9(2), 117–128 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sander, J.W., Sander, T.: On the kernel of the coprime graph of integers. Integers 9(5), 569–579 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sárközy, G.N.: Complete tripartite subgraphs in the coprime graph of integers. Discrete Math. 202(1–3), 227–238 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tripathi, A., Vijay, S.: A note on a theorem of Erdős & Gallai. Discrete Math. 265(1–3), 417–420 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Mathematics and Applied Computer ScienceUniversity of HildesheimHildesheimGermany

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