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Self-complementary hypergraphs

100%
Wojda A.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 2
217-224
EN
A k-uniform hypergraph H = (V;E) is called self-complementary if there is a permutation σ:V → V, called self-complementing, such that for every k-subset e of V, e ∈ E if and only if σ(e) ∉ E. In other words, H is isomorphic with $H' = (V; \binom{V}{k} - E)$. In the present paper, for every k, (1 ≤ k ≤ n), we give a characterization of self-complementig permutations of k-uniform self-complementary hypergraphs of the order n. This characterization implies the well known results for self-complementing permutations of graphs, given independently in the years 1962-1963 by Sachs and Ringel, and those obtained for 3-uniform hypergraphs by Kocay, for 4-uniform hypergraphs by Szymański, and for general (not uniform) hypergraphs by Zwonek.
2
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A note on k-uniform self-complementary hypergraphs of given order

80%
Szymański A., Wojda A.
Discussiones Mathematicae Graph Theory
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2009
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tom 29
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nr 1
199-202
EN
We prove that a k-uniform self-complementary hypergraph of order n exists, if and only if $\binom{n}{k}$ is even.
3
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The Existence of Quasi Regular and Bi-Regular Self-Complementary 3-Uniform Hypergraphs

70%
Kamble L. N., Deshpande C. M., Bam B. Y.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 2
419-426
EN
A k-uniform hypergraph H = (V ;E) is called self-complementary if there is a permutation σ : V → V , called a complementing permutation, such that for every k-subset e of V , e ∈ E if and only if σ(e) ∉ E. In other words, H is isomorphic with H′ = (V ; V(k) − E). In this paper we define a bi-regular hypergraph and prove that there exists a bi-regular self-complementary 3-uniform hypergraph on n vertices if and only if n is congruent to 0 or 2 modulo 4. We also prove that there exists a quasi regular self-complementary 3-uniform hypergraph on n vertices if and only if n is congruent to 0 modulo 4.
4
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Almost Self-Complementary 3-Uniform Hypergraphs

70%
Kamble L. N., Deshpande C. M., Bam B. Y.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
131-140
EN
It is known that self-complementary 3-uniform hypergraphs on n vertices exist if and only if n is congruent to 0, 1 or 2 modulo 4. In this paper we define an almost self-complementary 3-uniform hypergraph on n vertices and prove that it exists if and only if n is congruent to 3 modulo 4. The structure of corresponding complementing permutation is also analyzed. Further, we prove that there does not exist a regular almost self-complementary 3-uniform hypergraph on n vertices where n is congruent to 3 modulo 4, and it is proved that there exist a quasi regular almost self-complementary 3-uniform hypergraph on n vertices where n is congruent to 3 modulo 4.
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