Planar graphs with neither 5-cycles nor close 3-cycles are 3-colorable



In 2003, O. V. Borodin and A. Raspaud conjectured that every planar graph with the minimum distance between triangles, dΔ, at least 1 and without 5-cycles is 3-colorable. This Bordeaux conjecture (Brdx3CC) has common features with Havel's (1970) and Steinberg's (1976) 3-color problems. A relaxation of Brdx3CC with dΔ⩾4 was confirmed in 2003 by Borodin and Raspaud, whose result was improved to dΔ⩾3 by Borodin and A. N. Glebov (2004) and, independently, by B. Xu (2007). The purpose of this article is to make the penultimate step (dΔ⩾2) towards Brdx3CC. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 1–31, 2010