Almost t-complementary uniform hypergraphs

  • Shonda GosselinEmail author


An almost t-complementary k-hypergraph is a k-uniform hypergraph with vertex set V and edge set E for which there exists a permutation \(\theta \in Sym(V)\) such that the sets \(E, E^\theta , E^{\theta ^2}, \ldots , E^{\theta ^{t-1}}\) partition the set of all k-subsets of V minus one edge. Such a permutation \(\theta \) is called an almost (t, k)-complementing permutation. Almost t-complementary k-hypergraphs are a natural generalization of almost self-complementary graphs, which were previously studied by Clapham, Kamble et al., and Wojda. We prove that there exists an almost p-complementary k-hypergraph of order n whenever the base-p representation of k is a subsequence of the base-p representation of n, where p is prime.


Almost self-complementary hypergraph Uniform hypergraph Almost (t, k)-complementing permutation 

Mathematics Subject Classification

05C65 05E20 05C25 05C85 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Adamus, L., Orchel, B., Szymanski, A., Wojda, P., Zwonek, M.: A note on t-complementing permutations for graphs. Inform. Process. Lett. 110(2), 44–45 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bernaldez, J.M.: On \(k\)-complementing permutations of cyclically \(k\)-complementary graphs. Discrete Math. 151, 67–70 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Clapham, C.R.J.: Graphs self-complementary in \(K_n-e\). Discrete Math. 81, 229–235 (1990)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Colbourn, M.J., Colbourn, C.J.: Graph isomorphism and self-complementary graphs. SIGACT News 10(1), 25–29 (1978)CrossRefGoogle Scholar
  5. 5.
    Farrugia, A.: Self-complementary graphs and generalizations: a comprehensive reference manual. Master’s thesis, University of Malta (1999)Google Scholar
  6. 6.
    Gosselin, S.: Cyclically \(t\)-complementary uniform hypergraphs. Eur. J. Combin. 31, 1629–1636 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gosselin, S.: Generating self-complementary uniform hypergraphs. Discrete Math. 310, 1366–1372 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kamble, L.N., Deshpande, C.M., Bam, B.Y.: Almost self-complementary 3-uniform hypergraphs. Discuss. Math. Graph Theory 37, 131–140 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kummer, E.: Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. J. die reine angew. Math. 44, 93–146 (1852)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Suprunenko, D.A.: Self-complementary graphs. Cybernetica 21, 559–567 (1985)CrossRefGoogle Scholar
  11. 11.
    Szymański, A., Wojda, A.P.: Cyclic partitions of complete unifrom hypergraphs. Electron. J. Comb. 17, #R118, 1–12 (2010)Google Scholar
  12. 12.
    Szymański, A., Wojda, A.P.: Self-complementing permutations of \(k\)-uniform hypergraphs. Discrete Math. Theor. Comput. Sci. 11, 117–123 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Szymański, A., Wojda, A.P.: A note on \(k\)-uniform self-complementary hypergraphs of given order. Discuss. Math. Graph Theory 29, 199–202 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wojda, A.P.: Almost self-complementary uniform hypergraphs. Discuss. Math. Graph Theory 38, 607–610 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wojda, A.P.: Self-complementary hypergraphs. Discuss. Math. Graph Theory 26, 217–224 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of WinnipegWinnipegCanada

Personalised recommendations