### Abstract

Fix t > 1, a positive integer, and a = (a_{1},...,a_{t}) a vector of nonnegative integers. A t-coloring of the edges of a complete graph is called a-split if there exists a partition of the vertices into t sets V_{1},...,V_{t} such that every set of a_{i} + 1 vertices in V_{i} contains an edge of color i, for i = 1,...,t. We combine a theorem of Deza with Ramsey's theorem to prove that, for any fixed a, the family of a-split colorings is characterized by a finite list of forbidden induced subcolorings. A similar hypergraph version follows from our proofs. These results generalize previous work by Kézdy et al. (J. Combin. Theory Ser. A 73(2) (1996) 353) and Gyárfás (J. Combin. Theory Ser. A 81(2) (1998) 255). We also consider other notions of splitting.

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

Pages (from-to) | 415-421 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 257 |

Issue number | 2-3 |

Publication status | Published - Nov 28 2002 |

### Fingerprint

### Keywords

- Coloring
- Finite basis
- Ramsey theory

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*257*(2-3), 415-421.

**A finite basis characterization of α-split colorings.** / Gyárfás, A.; Kézdy, André E.; Lehel, K. Jeno.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 257, no. 2-3, pp. 415-421.

}

TY - JOUR

T1 - A finite basis characterization of α-split colorings

AU - Gyárfás, A.

AU - Kézdy, André E.

AU - Lehel, K. Jeno

PY - 2002/11/28

Y1 - 2002/11/28

N2 - Fix t > 1, a positive integer, and a = (a1,...,at) a vector of nonnegative integers. A t-coloring of the edges of a complete graph is called a-split if there exists a partition of the vertices into t sets V1,...,Vt such that every set of ai + 1 vertices in Vi contains an edge of color i, for i = 1,...,t. We combine a theorem of Deza with Ramsey's theorem to prove that, for any fixed a, the family of a-split colorings is characterized by a finite list of forbidden induced subcolorings. A similar hypergraph version follows from our proofs. These results generalize previous work by Kézdy et al. (J. Combin. Theory Ser. A 73(2) (1996) 353) and Gyárfás (J. Combin. Theory Ser. A 81(2) (1998) 255). We also consider other notions of splitting.

AB - Fix t > 1, a positive integer, and a = (a1,...,at) a vector of nonnegative integers. A t-coloring of the edges of a complete graph is called a-split if there exists a partition of the vertices into t sets V1,...,Vt such that every set of ai + 1 vertices in Vi contains an edge of color i, for i = 1,...,t. We combine a theorem of Deza with Ramsey's theorem to prove that, for any fixed a, the family of a-split colorings is characterized by a finite list of forbidden induced subcolorings. A similar hypergraph version follows from our proofs. These results generalize previous work by Kézdy et al. (J. Combin. Theory Ser. A 73(2) (1996) 353) and Gyárfás (J. Combin. Theory Ser. A 81(2) (1998) 255). We also consider other notions of splitting.

KW - Coloring

KW - Finite basis

KW - Ramsey theory

UR - http://www.scopus.com/inward/record.url?scp=32644436562&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=32644436562&partnerID=8YFLogxK

M3 - Article

VL - 257

SP - 415

EP - 421

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 2-3

ER -