### Abstract

Abstract A C-coloring of a hypergraph H = (X, ε) is a vertex coloring φ : X → ℕ such that each edge E ∈ ε has at least two vertices with a common color. The related parameter χ¯(H), called the upper chromatic number of H, is the maximum number of colors in a C-coloring of H. A hypertree is a hypergraph which has a host tree T such that each edge E ∈ ε induces a connected subgraph in T. Notations n and m stand for the number of vertices and edges, respectively, in a generic input hypergraph. We establish guaranteed polynomial-time approximation ratios for the difference n - χ¯(H), which is 2 + 2 ln(2m) on hypergraphs in general, and 1 + lnm on hypertrees. The latter ratio is essentially tight as we show that n - χ¯(H) cannot be approximated within (1 - ε)lnm on hypertrees (unless NP ⊆ DTIME(n^{O(loglogn)})). Furthermore, χ¯(H) does not have O(n^{1 - ε})-approximation and cannot be approximated within additive error o(n) on the class of hypertrees (unless P = NP).

Original language | English |
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Article number | 9927 |

Pages (from-to) | 1714-1721 |

Number of pages | 8 |

Journal | Discrete Mathematics |

Volume | 338 |

Issue number | 10 |

DOIs | |

Publication status | Published - May 31 2015 |

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### Keywords

- Approximation ratio
- C-coloring
- Hypergraph
- Hypertree
- Multiple hitting set
- Upper chromatic number

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*338*(10), 1714-1721. [9927]. https://doi.org/10.1016/j.disc.2014.08.007