### Abstract

The problem of coloring a graph with the minimum number of colors is well known to be NP-hard, even restricted to k-colorable graphs for constant k ≥ 3. On the other hand, it is known that random k-colorable graphs are easy to k-color. The algorithms for coloring random k-colorable graphs require fairly high edge densities, however. In this paper we present algorithms that color randomly generated k-colorable graphs for much lower edge densities than previous approaches. In addition, to study a wider variety of graph distributions, we also present a model of graphs generated by the semi-random source of Santha and Vazirani (M. Santha and U. V. Vazirani, J. Comput. System Sci.33 (1986), 75-87) that provides a smooth transition between the worst-case and random models. In this model, the graph is generated by a "noisy adversary"-an adversary whose decisions (whether or not to insert a particular edge) have some small (random) probability of being reversed. We show that even for quite low noise rates, semi-random k-colorable graphs can be optimally colored with high probability.

Original language | English (US) |
---|---|

Pages (from-to) | 204-234 |

Number of pages | 31 |

Journal | Journal of Algorithms |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - Sep 1995 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computational Mathematics
- Control and Optimization

### Cite this

*Journal of Algorithms*,

*19*(2), 204-234. https://doi.org/10.1006/jagm.1995.1034

**Coloring Random and Semi-Random k-Colorable Graphs.** / Blum, A.; Spencer, J.

Research output: Contribution to journal › Article

*Journal of Algorithms*, vol. 19, no. 2, pp. 204-234. https://doi.org/10.1006/jagm.1995.1034

}

TY - JOUR

T1 - Coloring Random and Semi-Random k-Colorable Graphs

AU - Blum, A.

AU - Spencer, J.

PY - 1995/9

Y1 - 1995/9

N2 - The problem of coloring a graph with the minimum number of colors is well known to be NP-hard, even restricted to k-colorable graphs for constant k ≥ 3. On the other hand, it is known that random k-colorable graphs are easy to k-color. The algorithms for coloring random k-colorable graphs require fairly high edge densities, however. In this paper we present algorithms that color randomly generated k-colorable graphs for much lower edge densities than previous approaches. In addition, to study a wider variety of graph distributions, we also present a model of graphs generated by the semi-random source of Santha and Vazirani (M. Santha and U. V. Vazirani, J. Comput. System Sci.33 (1986), 75-87) that provides a smooth transition between the worst-case and random models. In this model, the graph is generated by a "noisy adversary"-an adversary whose decisions (whether or not to insert a particular edge) have some small (random) probability of being reversed. We show that even for quite low noise rates, semi-random k-colorable graphs can be optimally colored with high probability.

AB - The problem of coloring a graph with the minimum number of colors is well known to be NP-hard, even restricted to k-colorable graphs for constant k ≥ 3. On the other hand, it is known that random k-colorable graphs are easy to k-color. The algorithms for coloring random k-colorable graphs require fairly high edge densities, however. In this paper we present algorithms that color randomly generated k-colorable graphs for much lower edge densities than previous approaches. In addition, to study a wider variety of graph distributions, we also present a model of graphs generated by the semi-random source of Santha and Vazirani (M. Santha and U. V. Vazirani, J. Comput. System Sci.33 (1986), 75-87) that provides a smooth transition between the worst-case and random models. In this model, the graph is generated by a "noisy adversary"-an adversary whose decisions (whether or not to insert a particular edge) have some small (random) probability of being reversed. We show that even for quite low noise rates, semi-random k-colorable graphs can be optimally colored with high probability.

UR - http://www.scopus.com/inward/record.url?scp=0001215301&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001215301&partnerID=8YFLogxK

U2 - 10.1006/jagm.1995.1034

DO - 10.1006/jagm.1995.1034

M3 - Article

AN - SCOPUS:0001215301

VL - 19

SP - 204

EP - 234

JO - Journal of Algorithms

JF - Journal of Algorithms

SN - 0196-6774

IS - 2

ER -