## Abstract

Coloring a k-colorable graph using k colors (k≥3) is a notoriously hard problem. Considering average case analysis allows for better results. In this work we consider the uniform distribution over k-colorable graphs with n vertices and exactly cn edges, c greater than some sufficiently large constant. We rigorously show that all proper k-colorings of most such graphs lie in a single "cluster", and agree on all but a small, though constant, portion of the vertices. We also describe a polynomial time algorithm that whp finds a proper k-coloring of such a random k-colorable graph, thus asserting that most such graphs are easy to color. This should be contrasted with the setting of very sparse random graphs (which are k-colorable whp), where experimental results show some regime of edge density to be difficult for many coloring heuristics.

Original language | English |
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Pages (from-to) | 523-565 |

Number of pages | 43 |

Journal | Theory of Computing Systems |

Volume | 46 |

Issue number | 3 |

DOIs | |

State | Published - 1 Apr 2010 |

Externally published | Yes |

## Keywords

- Average case analysis
- Random graphs
- Spectral analysis
- k-colorability

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics