Erdős–Pyber Theorem for Hypergraphs and Secret Sharing

László Csirmaz, Péter Ligeti, G. Tardos

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

A new, constructive proof with a small explicit constant is given to the Erdős–Pyber theorem which says that the edges of a graph on n vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most (Formula presented.) times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for graphs and for uniform hypergraphs. We show that these latter constants cannot be improved by more than a factor of 1.89 even for fractional covering by arbitrary complete multipartite subgraphs or subhypergraphs. In the case every vertex of the graph is connected to at least n-m other vertices, we prove the existence of a fractional covering of the edges by complete bipartite graphs such that every vertex is covered at most (Formula presented.) times, with only a slightly worse explicit constant. This result also generalizes to uniform hypergraphs. Our results give new improved bounds on the complexity of graph and uniform hypergraph based secret sharing schemes, and show the limits of the method at the same time.

Original languageEnglish
Pages (from-to)1335-1346
Number of pages12
JournalGraphs and Combinatorics
Volume31
Issue number5
DOIs
Publication statusPublished - May 16 2014

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Secret Sharing
Uniform Hypergraph
Hypergraph
Fractional
Graph in graph theory
Theorem
Subgraph
Covering
Vertex of a graph
Secret Sharing Scheme
Complete Bipartite Graph
Partitioning
Generalise
Arbitrary

Keywords

  • Bipartite graph
  • Graph covering
  • Partition cover number
  • Secret sharing
  • Uniform hypergraph

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Erdős–Pyber Theorem for Hypergraphs and Secret Sharing. / Csirmaz, László; Ligeti, Péter; Tardos, G.

In: Graphs and Combinatorics, Vol. 31, No. 5, 16.05.2014, p. 1335-1346.

Research output: Contribution to journalArticle

Csirmaz, László ; Ligeti, Péter ; Tardos, G. / Erdős–Pyber Theorem for Hypergraphs and Secret Sharing. In: Graphs and Combinatorics. 2014 ; Vol. 31, No. 5. pp. 1335-1346.
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