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 Mat. Sb., 2010, Volume 201, Number 4, Pages 137–160 (Mi msb7480)

The existence of panchromatic colourings for uniform hypergraphs

D. A. Shabanov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The well-known extremal problem concerning panchromatic colouring of hypergraphs is investigated. An $r$-colouring of the set of vertices of a hypergraph is said to be panchromatic if each link of the hypergraph is incident to vertices of all colours. The quantity $p(n,r)$ defined as the minimum number of links in an $n$-uniform hypergraph which admits no panchromatic $r$-colourings is studied. New lower and upper bounds for $p(n,r)$ are obtained, which improve earlier results for many of the relations between the parameters $n$ and $r$. In addition, a sufficient condition for the existence of a panchromatic $r$-colouring of an $n$-uniform hypergraph is obtained.
Bibliography: 18 items.

Keywords: hypergraph, panchromatic colouring, the method of random colouring.

DOI: https://doi.org/10.4213/sm7480

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English version:
Sbornik: Mathematics, 2010, 201:4, 607–630

Bibliographic databases:

UDC: 519.112.7+519.179.1
MSC: Primary 05C15; Secondary 05C65

Citation: D. A. Shabanov, “The existence of panchromatic colourings for uniform hypergraphs”, Mat. Sb., 201:4 (2010), 137–160; Sb. Math., 201:4 (2010), 607–630

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7480
• https://doi.org/10.4213/sm7480
• http://mi.mathnet.ru/eng/msb/v201/i4/p137

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Shabanov D.A., “On a generalization of Rubin's theorem”, J. Graph Theory, 67:3 (2011), 226–234
2. A. P. Rozovskaya, D. A. Shabanov, “Improvement of the Lower Bound in the Kostochka Problem of Panchromatic Coloring of a Hypergraph”, Math. Notes, 89:6 (2011), 903–906
3. A. M. Raigorodskii, D. A. Shabanov, “The Erdős–Hajnal problem of hypergraph colouring, its generalizations, and related problems”, Russian Math. Surveys, 66:5 (2011), 933–1002
4. A. P. Rozovskaya, D. A. Shabanov, “Extremal problems for panchromatic colourings of uniform hypergraphs”, Discrete Math. Appl., 22:2 (2012), 185–206
5. Kang R.J., “Improper Choosability and Property B”, J. Graph Theory, 73:3 (2013), 342–353
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