### Abstract

A Gallai-coloring of a complete graph is an edge coloring such that no triangle is colored with three distinct colors. Gallai-colorings occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper) or information theory. Gallai-colorings extend 2-colorings of the edges of complete graphs. They actually turn out to be close to 2-colorings-without being trivial extensions. Here, we give a method to extend some results on 2-colorings to Gallai-colorings, among them known and new, easy and difficult results. The method works for Gallai-extendible families that include, for example, double stars and graphs of diameter at most d for 2= d, or complete bipartite graphs. It follows that every Gallai-colored K_{n} contains a monochromatic double star with at least 3 n+1/4 vertices, a monochromatic complete bipartite graph on at least n/2 vertices, monochromatic subgraphs of diameter two with at least 3 n/4 vertices, etc. The generalizations are not automatic though, for instance, a Gallai-colored complete graph does not necessarily contain a monochromatic star on n/2 vertices. It turns out that the extension is possible for graph classes closed under a simple operation called equalization. We also investigate Ramsey numbers of graphs in Gallai-colorings with a given number of colors. For any graph H let RG(r,H) be the minimum m such that in every Gallai-coloring of K_{m} with r colors, there is a monochromatic copy of H. We show that for fixed H, RG (r,H) is exponential in r if H is not bipartite; linear in r if H is bipartite but not a star; constant (does not depend on r) if His a star (and we determine its value).

Original language | English |
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Pages (from-to) | 233-243 |

Number of pages | 11 |

Journal | Journal of Graph Theory |

Volume | 64 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 1 2010 |

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### Keywords

- Gallai coloring
- Ramsey

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*64*(3), 233-243. https://doi.org/10.1002/jgt.20452