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Acyclic 3-choosability of sparse graphs with girth at least 7. (English) Zbl 1213.05049
Summary: Every planar graph is known to be acyclically 7-choosable and is conjectured to be acyclically 5-choosable [O.V. Borodin, D. G. Fon-Der-Flaas, A. Kostocha, A. Raspaud and E. Sopena, J. Graph Theory 40, No.2, 83–90 (2002; Zbl 1004.05029)]. This conjecture, if proved, would imply both Borodin’s acyclic 5-color theorem [O. V. Borodin, Discrete Math. 25, 211–236 (1979; Zbl 0406.05031)] and Thomassen’s 5-choosability theorem [C. Thomassen, J. Comb. Theory, Ser. B 62, No.1, 180-181 (1994; Zbl 0805.05023)]. However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions have also been obtained for a planar graph to be acyclically 4- and 3-choosable.
We prove that each planar graph of girth at least 7 is acyclically 3-choosable. This is a common strengthening of the facts that such a graph is acyclically 3-colorable and that a planar graph of girth at least 8 is acyclically 3-choosable. More generally, we prove that every graph with girth at least 7 and maximum average degree less than \(\frac{14}5\) is acyclically 3-choosable.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
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