Bordeaux 3-color conjecture and 3-choosability

Abstract

A graph G = ( V , E ) is list L-colorable if for a given list assignment L = { L ( v ) : v ∈ V } , there exists a proper coloring c of G such that c ( v ) ∈ L ( v ) for all v ∈ V . If G is list L-colorable for every list assignment with | L ( v ) | ⩾ k for all v ∈ V , then G is said to be k-choosable. In this paper, we prove that (1) every planar graph either without 4- and 5-cycles, and without triangles at distance less than 4, or without 4-, 5- and 6-cycles, and without triangles at distance less than 3 is 3-choosable; (2) there exists a non-3-choosable planar graph without 4-cycles, 5-cycles, and intersecting triangles. These results have some consequences on the Bordeaux 3-color conjecture by Borodin and Raspaud [A sufficient condition for planar graphs to be 3-colorable. J. Combin. Theory Ser. B 88 (2003) 17–27].