Ergemlidze, Beka and Győri, Ervin and Methuku, Abhishek (2018) 3-uniform hypergraphs and linear cycles∗. SIAM JOURNAL ON DISCRETE MATHEMATICS, 32 (2). pp. 933-950. ISSN 0895-4801
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Abstract
Gyárfás, Gyori, and Simonovits [J. Comb., 7 (2016), pp. 205–216] proved that if a 3-uniform hypergraph with n vertices has no linear cycles, then its independence number α ≥ 2 5 n . The hypergraph consisting of vertex disjoint copies of a complete hypergraph K5 3 on five vertices shows that equality can hold. They asked whether this bound can be improved if we exclude K5 3 as a subhypergraph and whether such a hypergraph is 2-colorable. In this paper, we answer these questions affirmatively. Namely, we prove that if a 3-uniform linear-cycle-free hypergraph doesn’t contain K5 3 as a subhypergraph, then it is 2-colorable. This result clearly implies that its independence number α ≥ n 2 . We show that this bound is sharp. Gyárfás, Gyori, and Simonovits also proved that a linear-cycle-free 3-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free 3-uniform hypergraph has a vertex of degree at most n − 2 when n ≥ 10. © 2018 Society for Industrial and Applied Mathematics.
Item Type: | Article |
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Uncontrolled Keywords: | hypergraphs; Mathematical techniques; Graph theory; Combinatorial mathematics; Hyper graph; Independence number; Vertex disjoint; 3-uniform hypergraphs; Extremal hypergraphs; linear cycles; |
Subjects: | Q Science / természettudomány > QA Mathematics / matematika |
SWORD Depositor: | MTMT SWORD |
Depositing User: | MTMT SWORD |
Date Deposited: | 12 Jan 2019 12:03 |
Last Modified: | 12 Jan 2019 12:03 |
URI: | http://real.mtak.hu/id/eprint/89767 |
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