### Abstract

A class γ of graphs is χ-bounded if there exists a function f such that χ (G) ≤ f (ω (G)) for all graphs G ∈ γ, where χ denotes chromatic number and ω denotes clique number. Gyárfás and Sumner independently conjectured that, for any tree T, the class Forb (T), consisting of graphs that do not contain T as an induced subgraph, is χ-bounded. The first author and Penrice showed that this conjecture is true for any radius two tree. Here we use the work of several authors to show that the conjecture is true for radius three trees obtained from radius two trees by making exactly one subdivision in every edge adjacent to the root. These are the only trees with radius greater than two, other than subdivided stars, for which the conjecture is known to be true.

Original language | English (US) |
---|---|

Pages (from-to) | 571-581 |

Number of pages | 11 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2004 |

### Fingerprint

### Keywords

- Chromatic number
- Clique number
- Forbidden induced subgraph
- Radius three tree
- Template

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*SIAM Journal on Discrete Mathematics*,

*17*(4), 571-581. https://doi.org/10.1137/S0895480198339869

**Radius three trees in graphs with large chromatic number.** / Kierstead, Henry; Zhu, Yingxian.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 17, no. 4, pp. 571-581. https://doi.org/10.1137/S0895480198339869

}

TY - JOUR

T1 - Radius three trees in graphs with large chromatic number

AU - Kierstead, Henry

AU - Zhu, Yingxian

PY - 2004/4

Y1 - 2004/4

N2 - A class γ of graphs is χ-bounded if there exists a function f such that χ (G) ≤ f (ω (G)) for all graphs G ∈ γ, where χ denotes chromatic number and ω denotes clique number. Gyárfás and Sumner independently conjectured that, for any tree T, the class Forb (T), consisting of graphs that do not contain T as an induced subgraph, is χ-bounded. The first author and Penrice showed that this conjecture is true for any radius two tree. Here we use the work of several authors to show that the conjecture is true for radius three trees obtained from radius two trees by making exactly one subdivision in every edge adjacent to the root. These are the only trees with radius greater than two, other than subdivided stars, for which the conjecture is known to be true.

AB - A class γ of graphs is χ-bounded if there exists a function f such that χ (G) ≤ f (ω (G)) for all graphs G ∈ γ, where χ denotes chromatic number and ω denotes clique number. Gyárfás and Sumner independently conjectured that, for any tree T, the class Forb (T), consisting of graphs that do not contain T as an induced subgraph, is χ-bounded. The first author and Penrice showed that this conjecture is true for any radius two tree. Here we use the work of several authors to show that the conjecture is true for radius three trees obtained from radius two trees by making exactly one subdivision in every edge adjacent to the root. These are the only trees with radius greater than two, other than subdivided stars, for which the conjecture is known to be true.

KW - Chromatic number

KW - Clique number

KW - Forbidden induced subgraph

KW - Radius three tree

KW - Template

UR - http://www.scopus.com/inward/record.url?scp=9744257705&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=9744257705&partnerID=8YFLogxK

U2 - 10.1137/S0895480198339869

DO - 10.1137/S0895480198339869

M3 - Article

AN - SCOPUS:9744257705

VL - 17

SP - 571

EP - 581

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 4

ER -