### Abstract

For r≥2, α≥r-1 and k≥1, let c(r,α,k be the smallest integer c such that the vertex set of any non-trivial r-uniform k-edge-colored hypergraph H with α(H=α can be covered by c monochromatic connected components. Here α(H is the maximum cardinality of a subset A of vertices in H such that A does not contain any edges. An old conjecture of Ryser is equivalent to c(2,α,k=α(r-1 and a recent result of Z. Király states that c(r,r-1,k=⌈/kr⌉ for any r≥3. Here we make the first step to treat non-complete hypergraphs, showing that c(r,r,r=2 for r≥2 and c(r,r,r+1=3 for r≥3.

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

Article number | P2.33 |

Journal | Electronic Journal of Combinatorics |

Volume | 21 |

Issue number | 2 |

Publication status | Published - May 13 2014 |

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### Keywords

- Edge-coloring
- Graph theory
- Monochromatic component

### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*21*(2), [P2.33].

**A note on covering edge colored hypergraphs by monochromatic components.** / Fujita, Shinya; Furuya, Michitaka; Gyárfás, A.; Tóth, Ágnes.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 21, no. 2, P2.33.

}

TY - JOUR

T1 - A note on covering edge colored hypergraphs by monochromatic components

AU - Fujita, Shinya

AU - Furuya, Michitaka

AU - Gyárfás, A.

AU - Tóth, Ágnes

PY - 2014/5/13

Y1 - 2014/5/13

N2 - For r≥2, α≥r-1 and k≥1, let c(r,α,k be the smallest integer c such that the vertex set of any non-trivial r-uniform k-edge-colored hypergraph H with α(H=α can be covered by c monochromatic connected components. Here α(H is the maximum cardinality of a subset A of vertices in H such that A does not contain any edges. An old conjecture of Ryser is equivalent to c(2,α,k=α(r-1 and a recent result of Z. Király states that c(r,r-1,k=⌈/kr⌉ for any r≥3. Here we make the first step to treat non-complete hypergraphs, showing that c(r,r,r=2 for r≥2 and c(r,r,r+1=3 for r≥3.

AB - For r≥2, α≥r-1 and k≥1, let c(r,α,k be the smallest integer c such that the vertex set of any non-trivial r-uniform k-edge-colored hypergraph H with α(H=α can be covered by c monochromatic connected components. Here α(H is the maximum cardinality of a subset A of vertices in H such that A does not contain any edges. An old conjecture of Ryser is equivalent to c(2,α,k=α(r-1 and a recent result of Z. Király states that c(r,r-1,k=⌈/kr⌉ for any r≥3. Here we make the first step to treat non-complete hypergraphs, showing that c(r,r,r=2 for r≥2 and c(r,r,r+1=3 for r≥3.

KW - Edge-coloring

KW - Graph theory

KW - Monochromatic component

UR - http://www.scopus.com/inward/record.url?scp=84900555543&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84900555543&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84900555543

VL - 21

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2

M1 - P2.33

ER -