### Abstract

A graphGis called a (p,q)-split graph if its vertex set can be partitioned intoA,Bso that the order of the largest independent set inAis at mostpand the order of the largest complete subgraph inBis at mostq. Applying a well-known theorem of Erdos and Rado forΔ-systems, it is shown that for fixedp,q, (p,q)-split graphs can be characterized by excluding a finite set of forbidden subgraphs, called (p,q)-split critical graphs. The order of the largest (p,q)-split critical graph,f(p,q), relates to classical Ramsey numbersR(s,t) through the inequalities2F(F(R(p+2,q+2)))+1≥f(p,q)≥R(p+2,q+2)-1whereF(t) is the smallest number oft-element sets ensuring at+1-elementΔ-system. Apart fromf(1,1)=5, all values off(p,q) are unknown.

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

Pages (from-to) | 255-261 |

Number of pages | 7 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 81 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1998 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**Generalized Split Graphs and Ramsey Numbers.** / Gyárfás, A.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 81, no. 2, pp. 255-261. https://doi.org/10.1006/jcta.1997.2833

}

TY - JOUR

T1 - Generalized Split Graphs and Ramsey Numbers

AU - Gyárfás, A.

PY - 1998/2

Y1 - 1998/2

N2 - A graphGis called a (p,q)-split graph if its vertex set can be partitioned intoA,Bso that the order of the largest independent set inAis at mostpand the order of the largest complete subgraph inBis at mostq. Applying a well-known theorem of Erdos and Rado forΔ-systems, it is shown that for fixedp,q, (p,q)-split graphs can be characterized by excluding a finite set of forbidden subgraphs, called (p,q)-split critical graphs. The order of the largest (p,q)-split critical graph,f(p,q), relates to classical Ramsey numbersR(s,t) through the inequalities2F(F(R(p+2,q+2)))+1≥f(p,q)≥R(p+2,q+2)-1whereF(t) is the smallest number oft-element sets ensuring at+1-elementΔ-system. Apart fromf(1,1)=5, all values off(p,q) are unknown.

AB - A graphGis called a (p,q)-split graph if its vertex set can be partitioned intoA,Bso that the order of the largest independent set inAis at mostpand the order of the largest complete subgraph inBis at mostq. Applying a well-known theorem of Erdos and Rado forΔ-systems, it is shown that for fixedp,q, (p,q)-split graphs can be characterized by excluding a finite set of forbidden subgraphs, called (p,q)-split critical graphs. The order of the largest (p,q)-split critical graph,f(p,q), relates to classical Ramsey numbersR(s,t) through the inequalities2F(F(R(p+2,q+2)))+1≥f(p,q)≥R(p+2,q+2)-1whereF(t) is the smallest number oft-element sets ensuring at+1-elementΔ-system. Apart fromf(1,1)=5, all values off(p,q) are unknown.

UR - http://www.scopus.com/inward/record.url?scp=0040682060&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040682060&partnerID=8YFLogxK

U2 - 10.1006/jcta.1997.2833

DO - 10.1006/jcta.1997.2833

M3 - Article

AN - SCOPUS:0040682060

VL - 81

SP - 255

EP - 261

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -