### Abstract

Let G be a connected graph with n vertices and m edges (multiple edges allowed), and let k ≥ 2 be an integer. There is an algorithm with (optimal) running time of O(m) that finds (i) a bipartite subgraph of G with ≥ m/2 + (n − 1)/4 edges, (ii) a bipartite subgraph of G with ≥ m/2 + 3(n−1)/8 edges if G is triangle-free, (iii) a k-colourable subgraph of G with ≥ m − m/k + (n−1)/k + (k − 3)/2 edges if k ≥ 3 and G is not k-colorable. Infinite families of graphs show that each of those lower bounds on the worst-case performance are best possible (for every algorithm). Moreover, even if short cycles are excluded, the general lower bound of m − m/k cannot be replaced by m − m/k + εm for any fixed ε > 0; and it is NP-complete to decide whether a graph with m edges contains a k-colorable subgraph with more than m − m/k + εm edges, for any k ≥ 2 and ε> 0, ε <1/k.

Original language | English |
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Pages (from-to) | 201-210 |

Number of pages | 10 |

Journal | Combinatorics Probability and Computing |

Volume | 2 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1993 |

### ASJC Scopus subject areas

- Applied Mathematics
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics

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## Cite this

*Combinatorics Probability and Computing*,

*2*(2), 201-210. https://doi.org/10.1017/S0963548300000596