### Abstract

It is a well-known proposition that every graph of chromatic number larger than t contains every tree with t edges. The 'standard' reasoning is that such a graph must contain a subgraph of minimum degree at least t. Bohman, Frieze, and Mubayi noticed that, although this argument does not work for hypergraphs, it is still possible that the proposition holds for hypergraphs as well. Indeed, Loh recently proved that every uniform hypergraph of chromatic number larger than t contains every hypertree with t edges. Here we observe that the basic property of the well-known greedy algorithm immediately implies a much more general result (with a conceptually simpler proof): if the greedy algorithm colors the vertices of an r-uniform hypergraph with more than t colors then the hypergraph contains every r-uniform hypertree with t edges.

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

Pages (from-to) | 208-209 |

Number of pages | 2 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Feb 6 2011 |

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### Keywords

- Chromatic number
- Greedy coloring
- Subtrees in graphs

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*311*(2-3), 208-209. https://doi.org/10.1016/j.disc.2010.10.017

**Trees in greedy colorings of hypergraphs.** / Gyárfás, A.; Lehel, Jenö.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 311, no. 2-3, pp. 208-209. https://doi.org/10.1016/j.disc.2010.10.017

}

TY - JOUR

T1 - Trees in greedy colorings of hypergraphs

AU - Gyárfás, A.

AU - Lehel, Jenö

PY - 2011/2/6

Y1 - 2011/2/6

N2 - It is a well-known proposition that every graph of chromatic number larger than t contains every tree with t edges. The 'standard' reasoning is that such a graph must contain a subgraph of minimum degree at least t. Bohman, Frieze, and Mubayi noticed that, although this argument does not work for hypergraphs, it is still possible that the proposition holds for hypergraphs as well. Indeed, Loh recently proved that every uniform hypergraph of chromatic number larger than t contains every hypertree with t edges. Here we observe that the basic property of the well-known greedy algorithm immediately implies a much more general result (with a conceptually simpler proof): if the greedy algorithm colors the vertices of an r-uniform hypergraph with more than t colors then the hypergraph contains every r-uniform hypertree with t edges.

AB - It is a well-known proposition that every graph of chromatic number larger than t contains every tree with t edges. The 'standard' reasoning is that such a graph must contain a subgraph of minimum degree at least t. Bohman, Frieze, and Mubayi noticed that, although this argument does not work for hypergraphs, it is still possible that the proposition holds for hypergraphs as well. Indeed, Loh recently proved that every uniform hypergraph of chromatic number larger than t contains every hypertree with t edges. Here we observe that the basic property of the well-known greedy algorithm immediately implies a much more general result (with a conceptually simpler proof): if the greedy algorithm colors the vertices of an r-uniform hypergraph with more than t colors then the hypergraph contains every r-uniform hypertree with t edges.

KW - Chromatic number

KW - Greedy coloring

KW - Subtrees in graphs

UR - http://www.scopus.com/inward/record.url?scp=79953805642&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953805642&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2010.10.017

DO - 10.1016/j.disc.2010.10.017

M3 - Article

VL - 311

SP - 208

EP - 209

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 2-3

ER -