Acyclic 4-choosability of planar graphs

A proper vertex coloring of a graph G=(V,E)G=(V,E) is acyclic if GG contains no bicolored cycle. Given a list assignment L={L(v)∣v∈V}L={L(v)∣v∈V} of GG, we say GG is acyclically LL-list colorable if there exists a proper acyclic coloring ππ of GG such that π(v)∈L(v)π(v)∈L(v) for all v∈Vv∈V. If GG is acyclically LL-list colorable for any list assignment with |L(v)|≥k|L(v)|≥k for all v∈Vv∈V, then GG is acyclically kk-choosable. In this paper we prove that planar graphs without 4, 7, and 8-cycles are acyclically 4-choosable.