### 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 language | English |
---|---|

Pages (from-to) | 1335-1346 |

Number of pages | 12 |

Journal | Graphs and Combinatorics |

Volume | 31 |

Issue number | 5 |

DOIs | |

Publication status | Published - May 16 2014 |

### Fingerprint

### 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

*Graphs and Combinatorics*,

*31*(5), 1335-1346. https://doi.org/10.1007/s00373-014-1448-7

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

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 31, no. 5, pp. 1335-1346. https://doi.org/10.1007/s00373-014-1448-7

}

TY - JOUR

T1 - Erdős–Pyber Theorem for Hypergraphs and Secret Sharing

AU - Csirmaz, László

AU - Ligeti, Péter

AU - Tardos, G.

PY - 2014/5/16

Y1 - 2014/5/16

N2 - 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.

AB - 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.

KW - Bipartite graph

KW - Graph covering

KW - Partition cover number

KW - Secret sharing

KW - Uniform hypergraph

UR - http://www.scopus.com/inward/record.url?scp=84939771656&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939771656&partnerID=8YFLogxK

U2 - 10.1007/s00373-014-1448-7

DO - 10.1007/s00373-014-1448-7

M3 - Article

AN - SCOPUS:84939771656

VL - 31

SP - 1335

EP - 1346

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 5

ER -