## Abstract

For a fixed graph H, what is the smallest number of colors C such that there is a proper edge-coloring of the complete graph Kn with C colors containing no two vertex-disjoint color-isomorphic copies, or repeats, of H? We study this function and its generalization to more than two copies using a variety of combinatorial, probabilistic, and algebraic techniques. For example, we show that for any tree T there exists a constant c such that any proper edge-coloring of Kn with at most cn^{2} colors contains two repeats of T, while there are colorings with at most c^{\prime} n^{3}/^{2} colors for some absolute constant c^{\prime} containing no three repeats of any tree with at least two edges. We also show that for any graph H containing a cycle there exist k and c such that there is a proper edge-coloring of Kn with at most cn colors containing no k repeats of H, while for a tree T with m edges, a coloring with o(n^{(}m+1)^{/m}) colors contains \omega (1) repeats of T.

Original language | English |
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Pages (from-to) | 2249-2264 |

Number of pages | 16 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 35 |

Issue number | 3 |

DOIs | |

State | Published - 2021 |

Externally published | Yes |

## Keywords

- Coloring
- Extremal problems
- Graphs