Article in volume
Authors:
Title:
Extending Potočnik and Šajna's conditions on the existence of vertex-transitive self-complementary $k$-hypergraphs
PDFSource:
Discussiones Mathematicae Graph Theory 43(1) (2023) 225-231
Received: 2019-08-04 , Revised: 2020-09-02 , Accepted: 2020-09-02 , Available online: 2020-09-23 , https://doi.org/10.7151/dmgt.2360
Abstract:
Let $\ell$ be a positive integer, $k=2^\ell$ or $k=2^\ell+1$, and let $n$ be a
positive integer with $n \equiv 1$ (mod $2^{\ell+1}$). For a prime $p$,
$n_{(p)}$ denotes the largest integer $i$ such that $p^i$ divides $n$.
Potočnik and Šajna showed that if there exists a vertex-transitive
self-complementary $k$-hypergraph of order $n$, then for every prime $p$ we
have $p^{n_{(p)}} \equiv 1 \pmod {2^{\ell+1}}$. Here we extend their result
to a larger class of integers $k$.
Keywords:
vertex-transitive $k$-hypergraphs, self-complementary hypergraphs
References:
- R.A. Beezer, Sylow subgraphs in self-complementary vertex transitive graphs, Expo. Math. 24 (2006) 185–194.
https://doi.org/10.1016/j.exmath.2005.09.003 - M. Muzychuk, On Sylow subgraphs of vertex-transitive self-complementary graphs, Bull. Lond. Math. Soc. 31 (1999) 531–533.
https://doi.org/10.1112/S0024609399005925 - S. Gosselin, Vertex-transitive self-complementary uniform hypergraphs of prime order, Discrete Math. 310 (2010) 671–680.
https://doi.org/10.1016/j.disc.2009.08.011 - W.K. Nicholson, Introduction to Abstract Algebra (Wiley-Interscience, 2007).
- P. Potočnik and M. Šajna, Vetrex-transitive self-complementary uniform hypergraphs, European J. Combin. 28 (2009) 327–337.
https://doi.org/10.1016/j.ejc.2007.08.003 - S.B. Rao, On regular and strongly-regular self-complementary graphs, Discrete Math. 54 (1985) 73–82.
https://doi.org/10.1016/0012-365X(85)90063-9 - A. Szymański and A.P. Wojda, Self-complementing permutations of $k$-uniform hypergraphs, Discrete Math. Theor. Comput. Sci. 11 (2009) 117–124.
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