### Abstract

For any pair of graphs G and H, both the size Ramsey number r(G, H) and the restricted size Ramsey number r
^{∗}
(G, H) are bounded above by the size of the complete graph with order equals to the Ramsey number r(G, H), and bounded below by e(G) + e(H) − 1. Moreover, trivially, r(G, H) ≤ r
^{∗}
(G, H). When introducing the size Ramsey number for graph, Erdös et al. (1978) asked two questions; (1) Do there exist graphs G and H such that r(G, H) attains the upper bound? and (2) Do there exist graphs G and H such that r(G, H) is significantly less than the upper bound? In this paper we consider the restricted size Ramsey number r
^{∗}
(G, H). We answer both questions above for r
^{∗}
(G, H) when G = P
_{3}
and H is a connected graph.

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

Pages (from-to) | 741-755 |

Number of pages | 15 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 39 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Keywords

- Connected graph
- Path
- Restricted size ramsey number
- Star

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## Cite this

_{3}.

*Discussiones Mathematicae - Graph Theory*,

*39*(2), 741-755. https://doi.org/10.7151/dmgt.2188