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Mat. Zametki, 1997, Volume 61, Issue 6, Pages 873–883 (Mi mz1571)  

Estimates of the independence number of a hypergraph and the Ryser conjecture

V. E. Tarakanov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Consider a hypergraph $H$ with $n$ vertices and $m$ edges. Suppose that its edges contain at most $r$ vertices, $r\ge2$, and the degrees of its vertices do not exceed $\delta\ge2$. Let $\tau(H)$ be the transversal number of the graph $H$ and $\mu(H)$ be its independence number, i.e., the maximal number of pairwise nonintersecting edges containing $r$ vertices. We strengthen the known estimate $\mu\ge(\delta n-(r-1)m)/(\delta r-r+1)$ for the case in which $H$ is a pseudograph and the maximal degree of its vertices equals $\Delta$, $2\delta-2\ge\Delta$ (there is a similar result for graphs). Then we use the sharpened estimate to prove the well known Ryser conjecture on $r$-partite $r$-uniform hypergraphs $H$: this conjecture states that $\tau(H)\le(r-1)\mu(H)$, and we prove it for $\delta$-regular $H$, where $2\le\delta\le r-1$.

DOI: https://doi.org/10.4213/mzm1571

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English version:
Mathematical Notes, 1997, 61:6, 731–738

Bibliographic databases:

Document Type: Article
UDC: 519.17
Received: 19.09.1996

Citation: V. E. Tarakanov, “Estimates of the independence number of a hypergraph and the Ryser conjecture”, Mat. Zametki, 61:6 (1997), 873–883; Math. Notes, 61:6 (1997), 731–738

Citation in format AMSBIB
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\paper Estimates of the independence number of a hypergraph and the Ryser conjecture
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\yr 1997
\vol 61
\issue 6
\pages 873--883
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\jour Math. Notes
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\vol 61
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\pages 731--738
\crossref{https://doi.org/10.1007/BF02361215}
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