Acyclic 5-choosability of planar graphs without adjacent short cycles

Abstract

The conjecture on acyclic 5-choosability of planar graphs [Borodin et al., 2002] as yet has been verified only for several restricted classes of graphs. None of these classes allows 4-cycles. We prove that a planar graph is acyclically 5-choosable if it does not contain an i-cycle adjacent to a j-cycle where 3⩽j⩽5 if i = 3 and 4⩽j⩽6 if i = 4. This result absorbs most of the previous work in this direction. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:169-176, 2011