### Abstract

It is shown that if G and H are star-forests with no single edge stars, then (G, H) is Ramsey-finite if and only if both G and H are single stars with an odd number of edges. Further (S_{m} ∪kS_{1}, S_{n} ∪tS_{1}) is Ramsey-finite when m and n are odd, where S_{i} denotes a star with i edges. In general, for G and H star-forests, (G∪kS_{1}, H∪tS_{1}) can be shown to be Ramsey-finite or Ramsey-infinite depending on the choice of G, H, k, and l with the general case unsettled. This disproves the conjecture given in [2] where it is suggested that the pair of graphs (L, M) is Ramsey-finite if and only if (1) either L or M is a matching, or (2) both L and M are star-forests of the type S_{m} ∪kS_{1}, m odd and k ≥ 0.

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

Pages (from-to) | 227-237 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 33 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1981 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*33*(3), 227-237. https://doi.org/10.1016/0012-365X(81)90266-1

**Ramsey-minimal graphs for star-forests.** / Burr, Stefan A.; Erdős, P.; Faudree, R. J.; Rousseau, C. C.; Schelp, R. H.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 33, no. 3, pp. 227-237. https://doi.org/10.1016/0012-365X(81)90266-1

}

TY - JOUR

T1 - Ramsey-minimal graphs for star-forests

AU - Burr, Stefan A.

AU - Erdős, P.

AU - Faudree, R. J.

AU - Rousseau, C. C.

AU - Schelp, R. H.

PY - 1981

Y1 - 1981

N2 - It is shown that if G and H are star-forests with no single edge stars, then (G, H) is Ramsey-finite if and only if both G and H are single stars with an odd number of edges. Further (Sm ∪kS1, Sn ∪tS1) is Ramsey-finite when m and n are odd, where Si denotes a star with i edges. In general, for G and H star-forests, (G∪kS1, H∪tS1) can be shown to be Ramsey-finite or Ramsey-infinite depending on the choice of G, H, k, and l with the general case unsettled. This disproves the conjecture given in [2] where it is suggested that the pair of graphs (L, M) is Ramsey-finite if and only if (1) either L or M is a matching, or (2) both L and M are star-forests of the type Sm ∪kS1, m odd and k ≥ 0.

AB - It is shown that if G and H are star-forests with no single edge stars, then (G, H) is Ramsey-finite if and only if both G and H are single stars with an odd number of edges. Further (Sm ∪kS1, Sn ∪tS1) is Ramsey-finite when m and n are odd, where Si denotes a star with i edges. In general, for G and H star-forests, (G∪kS1, H∪tS1) can be shown to be Ramsey-finite or Ramsey-infinite depending on the choice of G, H, k, and l with the general case unsettled. This disproves the conjecture given in [2] where it is suggested that the pair of graphs (L, M) is Ramsey-finite if and only if (1) either L or M is a matching, or (2) both L and M are star-forests of the type Sm ∪kS1, m odd and k ≥ 0.

UR - http://www.scopus.com/inward/record.url?scp=4344713915&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344713915&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(81)90266-1

DO - 10.1016/0012-365X(81)90266-1

M3 - Article

AN - SCOPUS:4344713915

VL - 33

SP - 227

EP - 237

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 3

ER -