### 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. A basic property of Gallai-colorings with at least three colors is that at least one of the color classes must span a disconnected graph. We are interested here in whether this or a similar property remains true if we consider colorings that do not contain a rainbow copy of a fixed graph F. We show that such graphs F are very close to bipartite graphs, namely, they can be made bipartite by the removal of at most one edge. We also extend Gallai's property for two infinite families and show that it also holds when F is a path with at most six vertices.

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

Pages (from-to) | 104-114 |

Number of pages | 11 |

Journal | Journal of Graph Theory |

Volume | 74 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 2013 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*74*(1), 104-114. https://doi.org/10.1002/jgt.21694

**Disconnected colors in generalized Gallai-colorings.** / Fujita, Shinya; Gyárfás, A.; Magnant, Colton; Seress, Ákos.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 74, no. 1, pp. 104-114. https://doi.org/10.1002/jgt.21694

}

TY - JOUR

T1 - Disconnected colors in generalized Gallai-colorings

AU - Fujita, Shinya

AU - Gyárfás, A.

AU - Magnant, Colton

AU - Seress, Ákos

PY - 2013/9

Y1 - 2013/9

N2 - 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. A basic property of Gallai-colorings with at least three colors is that at least one of the color classes must span a disconnected graph. We are interested here in whether this or a similar property remains true if we consider colorings that do not contain a rainbow copy of a fixed graph F. We show that such graphs F are very close to bipartite graphs, namely, they can be made bipartite by the removal of at most one edge. We also extend Gallai's property for two infinite families and show that it also holds when F is a path with at most six vertices.

AB - 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. A basic property of Gallai-colorings with at least three colors is that at least one of the color classes must span a disconnected graph. We are interested here in whether this or a similar property remains true if we consider colorings that do not contain a rainbow copy of a fixed graph F. We show that such graphs F are very close to bipartite graphs, namely, they can be made bipartite by the removal of at most one edge. We also extend Gallai's property for two infinite families and show that it also holds when F is a path with at most six vertices.

UR - http://www.scopus.com/inward/record.url?scp=84879889324&partnerID=8YFLogxK

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U2 - 10.1002/jgt.21694

DO - 10.1002/jgt.21694

M3 - Article

VL - 74

SP - 104

EP - 114

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -