Almost t-complementary uniform hypergraphs


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.

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  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)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bernaldez, J.M.: On \(k\)-complementing permutations of cyclically \(k\)-complementary graphs. Discrete Math. 151, 67–70 (1996)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Clapham, C.R.J.: Graphs self-complementary in \(K_n-e\). Discrete Math. 81, 229–235 (1990)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Colbourn, M.J., Colbourn, C.J.: Graph isomorphism and self-complementary graphs. SIGACT News 10(1), 25–29 (1978)

    Article  Google Scholar 

  5. 5.

    Farrugia, A.: Self-complementary graphs and generalizations: a comprehensive reference manual. Master’s thesis, University of Malta (1999)

  6. 6.

    Gosselin, S.: Cyclically \(t\)-complementary uniform hypergraphs. Eur. J. Combin. 31, 1629–1636 (2010)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Gosselin, S.: Generating self-complementary uniform hypergraphs. Discrete Math. 310, 1366–1372 (2010)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kummer, E.: Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. J. die reine angew. Math. 44, 93–146 (1852)

    MathSciNet  Google Scholar 

  10. 10.

    Suprunenko, D.A.: Self-complementary graphs. Cybernetica 21, 559–567 (1985)

    Article  Google Scholar 

  11. 11.

    Szymański, A., Wojda, A.P.: Cyclic partitions of complete unifrom hypergraphs. Electron. J. Comb. 17, #R118, 1–12 (2010)

  12. 12.

    Szymański, A., Wojda, A.P.: Self-complementing permutations of \(k\)-uniform hypergraphs. Discrete Math. Theor. Comput. Sci. 11, 117–123 (2009)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Wojda, A.P.: Almost self-complementary uniform hypergraphs. Discuss. Math. Graph Theory 38, 607–610 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Wojda, A.P.: Self-complementary hypergraphs. Discuss. Math. Graph Theory 26, 217–224 (2006)

    MathSciNet  Article  Google Scholar 

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Correspondence to Shonda Gosselin.

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Gosselin, S. Almost t-complementary uniform hypergraphs. Aequat. Math. 93, 1177–1182 (2019).

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Mathematics Subject Classification

  • 05C65
  • 05E20
  • 05C25
  • 05C85


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