### Abstract

A subset of the vertices in a hypergraph is a cover if it intersects every edge. Let τ(H) denote the cardinality of a minimum cover in the hypergraph H, and let us denote by g(n) the maximum of τ(H) taken over all hypergraphs H with n vertices and with no two hyperedges of the same size. We show thatg(n)k edges with cardinality i for all i=1, 2, ..., n. We prove that g(n, C, k)(k+1)/(k+2).These results have an interesting graph-theoretic application. For a family F of graphs, let T(n, F, r) denote the maximum possible number of edges in a graph with n vertices, which contains each member of F at most r-1 times. T(n, F, 1)=T(n, F) is the classical Turán number. Using the results above, we can compute a non-trivial upper bound for T(n, F, r) for many interesting graph families.

Original language | English |
---|---|

Pages (from-to) | 270-284 |

Number of pages | 15 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 82 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 2001 |

### Fingerprint

### Keywords

- Hypergraph; covering; cycles

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series B*,

*82*(2), 270-284. https://doi.org/10.1006/jctb.2001.2037

**Covering non-uniform hypergraphs.** / Boros, Endre; Caro, Yair; Füredi, Z.; Yuster, Raphael.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 82, no. 2, pp. 270-284. https://doi.org/10.1006/jctb.2001.2037

}

TY - JOUR

T1 - Covering non-uniform hypergraphs

AU - Boros, Endre

AU - Caro, Yair

AU - Füredi, Z.

AU - Yuster, Raphael

PY - 2001/7

Y1 - 2001/7

N2 - A subset of the vertices in a hypergraph is a cover if it intersects every edge. Let τ(H) denote the cardinality of a minimum cover in the hypergraph H, and let us denote by g(n) the maximum of τ(H) taken over all hypergraphs H with n vertices and with no two hyperedges of the same size. We show thatg(n)k edges with cardinality i for all i=1, 2, ..., n. We prove that g(n, C, k)(k+1)/(k+2).These results have an interesting graph-theoretic application. For a family F of graphs, let T(n, F, r) denote the maximum possible number of edges in a graph with n vertices, which contains each member of F at most r-1 times. T(n, F, 1)=T(n, F) is the classical Turán number. Using the results above, we can compute a non-trivial upper bound for T(n, F, r) for many interesting graph families.

AB - A subset of the vertices in a hypergraph is a cover if it intersects every edge. Let τ(H) denote the cardinality of a minimum cover in the hypergraph H, and let us denote by g(n) the maximum of τ(H) taken over all hypergraphs H with n vertices and with no two hyperedges of the same size. We show thatg(n)k edges with cardinality i for all i=1, 2, ..., n. We prove that g(n, C, k)(k+1)/(k+2).These results have an interesting graph-theoretic application. For a family F of graphs, let T(n, F, r) denote the maximum possible number of edges in a graph with n vertices, which contains each member of F at most r-1 times. T(n, F, 1)=T(n, F) is the classical Turán number. Using the results above, we can compute a non-trivial upper bound for T(n, F, r) for many interesting graph families.

KW - Hypergraph; covering; cycles

UR - http://www.scopus.com/inward/record.url?scp=0035402510&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035402510&partnerID=8YFLogxK

U2 - 10.1006/jctb.2001.2037

DO - 10.1006/jctb.2001.2037

M3 - Article

AN - SCOPUS:0035402510

VL - 82

SP - 270

EP - 284

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -