### Abstract

For a vertex v of an edge-colored graph, the color degree of v is the number of colors appeared in edges incident with v. An edge-colored graph is called properly colored if no two adjacent edges have the same color. In this paper, we prove that if the minimum color degree sum of two adjacent vertices of an edge-colored connected graph G is at least |G|, then G has a properly colored spanning tree. This is a generalization of the result proved by Cheng, Kano and Wang. We also show the sharpness of this lower bound of the color degree sum.

Original language | English |
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Article number | 112042 |

Journal | Discrete Mathematics |

Volume | 343 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2020 Nov |

### Keywords

- Edge-colored graph
- Properly colored
- Rainbow
- Spanning tree

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Kano, M., Maezawa, S. I., Ota, K., Tsugaki, M., & Yashima, T. (2020). Color degree sum conditions for properly colored spanning trees in edge-colored graphs.

*Discrete Mathematics*,*343*(11), [112042]. https://doi.org/10.1016/j.disc.2020.112042