### Abstract

Gallai-colorings of complete graphs-edge colorings such that no triangle is colored with three distinct colors-occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper), information theory and the theory of perfect graphs. We extend here Gallai-colorings to non-complete graphs and study the analogue of a basic result-any Gallai-colored complete graph has a monochromatic spanning tree-in this more general setting. We show that edge colorings of a graph H without multicolored triangles contain monochromatic connected subgraphs with at least (α (H)^{2} + α (H) - 1)^{- 1} | V (H) | vertices, where α (H) is the independence number of H. In general, we show that if the edges of an r-uniform hypergraph H are colored so that there is no multicolored copy of a fixed F then there is a monochromatic connected subhypergraph H_{1} ⊆ H such that | V (H_{1}) | ≥ c | V (H) | where c depends only on F, r, and α (H).

Original language | English |
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Pages (from-to) | 977-980 |

Number of pages | 4 |

Journal | Discrete Mathematics |

Volume | 310 |

Issue number | 5 |

DOIs | |

Publication status | Published - Mar 6 2010 |

### Keywords

- Gallai colorings
- Monochromatic connected components

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*310*(5), 977-980. https://doi.org/10.1016/j.disc.2009.10.013