### Abstract

Let G be an edge-colored graph and v a vertex of G. The color degree of v is the number of colors appearing on the edges incident to v. A rainbow triangle in G is one in which all edges have distinct colors. In this paper, we first prove that an edge-colored graph on n vertices contains a rainbow triangle if the color degree sum of every two adjacent vertices is at least n+1n+1 . Afterwards, we characterize the edge-colored graphs on n vertices containing no rainbow triangles but satisfying that each pair of adjacent vertices has color degree sum at least n.

Original language | English |
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Pages (from-to) | 2001-2008 |

Journal | Graphs and combinatorics |

Volume | 32 |

Issue number | 5 |

DOIs | |

Publication status | Published - Sep 2016 |

### Keywords

- Edge-colored graphs
- Color degree
- Rainbow triangles

## Cite this

Li, R., Ning, B., & Zhang, S. (2016). Color Degree Sum Conditions for Rainbow Triangles in Edge-Colored Graphs.

*Graphs and combinatorics*,*32*(5), 2001-2008. https://doi.org/10.1007/s00373-016-1690-2