Planar graphs without 4- and 5-cycles are acyclically 4-choosable

• Authors: Min Chen;André Raspaud
• Venue: Discrete Applied Mathematics
• Year: 2013

Let G=(V,E) be a graph. A proper vertex coloring of G is acyclic if G contains no bicolored cycle. Namely, every cycle of G must be colored with at least three colors. G is acyclically L-list colorable if for a given list assignment L={L(v):v@?V}, there exists a proper acyclic coloring @p of G such that @p(v)@?L(v) for all v@?V. If G is acyclically L-list colorable for any list assignment with |L(v)|=k for all v@?V, then G is acyclically k-choosable. In 1976, Steinberg Jensen and Toft (1995) [20] conjectured that every planar graph without 4- and 5-cycles is 3-colorable. This conjecture cannot be improved to 3-choosability basing on the examples given by Voigt (2007) [30] and, independently, by Montassier (2005) [24]. In this paper, we prove that planar graphs without 4- and 5-cycles are acyclically 4-choosable. This result (obtained independently by Borodin and Ivanova (2012) [9]) is also a new approach to the conjecture proposed by Montassier et al. (2006) in [27], which says that every planar graph without 4-cycles is acyclically 4-choosable.