### Abstract

Given graphs G and H, a coloring of E(G) is called an (H, q)-coloring if the edges of every copy of H ⊆ G together receive at least q colors. Let r(G, H, q) denote the minimum number of colors in an (H, q)-coloring of G. We determine, for fixed p, the smallest q for which r(K_{n,n}, K_{p,p}, q) is linear in n, the smallest q for which it is quadratic in n. We also determine the smallest q for which r(K_{n,n}, K_{p,p}, q) =n^{2} - O(n), and the smallest q for which r(K_{n,n}, K_{p,p}, q) = n^{2} - O(1). Our results include showing that r(K_{n,n}, K_{2, t+1}, 2) and r(K_{n}, K_{2, t+1}, 2) are both (1 + 0(1)) √n/t as n → ∞, thereby proving a special case of a conjecture of Chung and Graham. Finally, we determine the exact value of r(K_{n,n}, K_{3,3}, 8), and prove that 2n/3≤r(K_{n,n},C_{4}, 3)≤n + 1. Several problems remain open.

Original language | English |
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Pages (from-to) | 66-86 |

Number of pages | 21 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 79 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 2000 |

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### Keywords

- Algebraic constructions
- Edge-coloring of bipartite graphs
- Projective planes
- Ramsey theory

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series B*,

*79*(1), 66-86. https://doi.org/10.1006/jctb.1999.1948