In 1970, Havel asked if each planar graph with the minimum distance, d ∇;, between triangles large enough is 3-colorable. There are 4-chromatic planar graphs with d ∇;=3 (Aksenov, Mel'nikov, and Steinberg, 1980). The first result in the positive direction of Havel's problem was made in 2003 by Borodin and Raspaud, who proved that every planar graph with d ∇;≥4 and no 5-cycles is 3-colorable.Recently, Havel's problem was solved by Dvořák, Král' and Thomas in the positive, which means that there exists a constant d such that each planar graph with d ∇;≥d is 3-colorable. (As far as we can judge, this d is very large.)We conjecture that the strongest possible version of Havel's problem (SVHP) is true: every planar graph with d ∇;≥4 is 3-colorable. In this paper we prove that each planar graph with d ∇;≥4 and without 5-cycles adjacent to triangles is 3-colorable. The readers are invited to prove a stronger theorem: every planar graph with d ∇;≥4 and without 4-cycles adjacent to triangles is 3-colorable, which could possibly open way to proving SVHP. © 2012 Elsevier Inc.
Borodin, O. V., Glebov, A. N., & Jensen, T. R. (2012). A step towards the strong version of Havel’s three color conjecture. Journal of Combinatorial Theory. Series B, 102(6), 1295–1320. https://doi.org/10.1016/j.jctb.2012.08.001