Journal of Computer and System Sciences 67 (2003) 441471
Almost all graphs with average degree 4 are 3-colorable
and Cristopher Mooreb,c,Ã,1
Microsoft Research, Redmond, Washington, USA
Computer Science Department, University of New Mexico, Farris Engineering Center, Albuquerque, NM 87131, USA
Santa Fe Institute, Santa Fe, NM, USA
Received 14 July 2002; revised 28 December 2002
We analyze a randomized version of the Brelaz heuristic on sparse random graphs. We prove that almost
all graphs with average degree dp4:03; i.e., Gðn; p ¼ d=nÞ; are 3-colorable and that a constant fraction of
all 4-regular graphs are 3-colorable.
r 2003 Elsevier Inc. All rights reserved.
Keywords: Random graphs; Regular random graphs; Graph coloring; Threshold phenomena
Let Gðn; pÞ be a random graph on n vertices where each edge appears independently of all