## Abstract

A ρ-mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most ρ. For a graph H and for p > 1, the mean Ramsey-Turán number RT(n, H, ρ-mean) is the maximum number of edges a ρ-mean colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. It is conjectured that RT(n, Km, 2 - mean) = RT(n, K_{m}, 2) where RT(n, H, k) is the maximum number of edges a k edge-colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. We prove the conjecture holds for K_{3}. We also prove that RT(n, H,ρ- mean) > RT(n, K_{χ(H)}, p - mean) + o(n^{2}). This result is tight for graphs H whose clique number equals their chromatic number. In particular, we get that if H is a 3-chromatic graph having a triangle then RT(n, H, 2- mean) = RT(n, K_{3}, 2 - mean) + o(n^{2}) = RT(n. K^{3}. 2) + o(n^{2}) = 0.4n^{2}(1 + o(1)).

Original language | English |
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Pages (from-to) | 126-134 |

Number of pages | 9 |

Journal | Journal of Graph Theory |

Volume | 53 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2006 |

## Keywords

- Coloring
- Ramsey numbers
- Turan numbers

## ASJC Scopus subject areas

- Geometry and Topology