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Planar graphs without 4- and 5-cycles are acyclically 4-choosable. (English) Zbl 1262.05029
Summary: 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 $$L=\{ L(v) : v \in V\}$$, there exists a proper acyclic coloring $$\pi$$ of $$G$$ such that $$\pi (v)\in L(v)$$ for all $$v\in V$$ . If $$G$$ is acyclically $$L$$-list colorable for any list assignment with $$|L(v)|\geq k$$ for all $$v\in V$$, then $$G$$ is acyclically $$k$$-choosable.
T. R. Jensen and B. Toft [Graph coloring problems. New York, NY: John Wiley & Sons (1995; Zbl 0855.05054)] 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 M. Voigt [Discrete Math. 307, No. 7–8, 1013–1015 (2007; Zbl 1112.05041)] and, independently, by M. Montassier [“A smaller not 3-choosable planar graph without 4- and 5-cycles”. Research Report RR-1361-05 (2005)].
In this paper, we prove that planar graphs without 4- and 5-cycles are acyclically 4-choosable. This result (obtained independently by O. V. Borodin and A. O. Ivanova [J. Graph Theory 72, No. 3–4, 374–397 (2013; Zbl 1261.05013)]) is also a new approach to the conjecture proposed by M. Montassier et al. [Algorithms and Combinatorics 26, 473–491 (2006; Zbl 1120.05034)], which says that every planar graph without 4-cycles is acyclically 4-choosable.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C15 Coloring of graphs and hypergraphs 05C38 Paths and cycles
##### Keywords:
planar graphs; acyclic coloring; choosability; cycle
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##### References:
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