Almost All Graphs of Degree 4 are 3-colorable

Dimitris Achlioptas and Cristopher Moore

Working Papers from Santa Fe Institute

Abstract: The technique of approximating the mean path of Markov chains by differential equations has proved to be a useful tool in analyzing the performance of heuristics on random graph instances. However, only a small family of algorithms can currently be analyzed by this method, due to the need to maintain uniform randomness within the original state space. Here, we significantly expand the range of the differential equation technique, by showing how it can be generalized to handle heuristics that give priority to high- or low-degree vertices. In particular, we focus on 3-coloring and analyze a "smoothed" version of the practically successful Brelaz heuristic. This allows to prove that almost all graphs with average degree $d$, i.e. $G(n,p=d/n)$, are 3-colorable for $d \leq 4.03$, and that almost all 4-regular graphs are 3-colorable. This improves over the previous lower bound of $3.847$ on the 3-colorability threshold for $G(n,p=d/n)$ and gives the first non-trivial result on the colorability of random regular graphs. In fact, our methods can be used to deal with "arbitrary" sparse degree distributions and in conjunction with general graph algorithms that have a preference for high- or low-degree vertices.

Keywords: Random graphs; graph coloring; analysis of algorithms; computational complexity; heuristics; phase transitions in NP-complete problems (search for similar items in EconPapers)
Date: 2001-11
References: View complete reference list from CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wop:safiwp:01-11-070

Access Statistics for this paper

More papers in Working Papers from Santa Fe Institute Contact information at EDIRC.
Bibliographic data for series maintained by Thomas Krichel ().