### Abstract

The chromatic capacityχ_{cap} (G) of a graph G is the largest k for which there exists a k-coloring of the edges of G such that, for every coloring of the vertices of G with the same colors, some edge is colored the same as both its vertices. We prove that there is an unbounded function f : N → N such that χ_{cap} (G) ≥ f (χ (G)) for almost every graph G, where χ denotes the chromatic number. We show that for any positive integers n and k with k ≤ n / 2 there exists a graph G with χ (G) = n and χ_{cap} (G) = n - k, extending a result of Greene. We obtain bounds on χ_{cap} (K_{n}^{r}) that are tight as r → ∞, where K_{n}^{r} is the complete n-partite graph with r vertices in each part. Finally, for any positive integers p and q we construct a graph G with χ_{cap} (G) + 1 = χ (G) = p that contains no odd cycles of length less than q.

Original language | English (US) |
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Pages (from-to) | 2134-2148 |

Number of pages | 15 |

Journal | Discrete Mathematics |

Volume | 308 |

Issue number | 11 |

DOIs | |

State | Published - Jun 6 2008 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*308*(11), 2134-2148. https://doi.org/10.1016/j.disc.2006.10.021