Contract grant sponsor: Russian Foundation for Basic Research; Contract grant numbers: 09-01-00244 and 08-01-00673. The first author was partly supported by the Ministry of education and science of the Russian Federation (contract number 14.740.11.0868).
Acyclic 4-Choosability of Planar Graphs with No 4- and 5-Cycles
Article first published online: 29 MAR 2012
© 2012 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 72, Issue 4, pages 374–397, April 2013
How to Cite
Borodin, O. V. and Ivanova, A. O. (2013), Acyclic 4-Choosability of Planar Graphs with No 4- and 5-Cycles. J. Graph Theory, 72: 374–397. doi: 10.1002/jgt.21647
- Issue published online: 8 FEB 2013
- Article first published online: 29 MAR 2012
- Manuscript Revised: 8 SEP 2011
- Manuscript Received: 7 MAY 2010
- acyclic coloring;
- planar graph;
- forbidden cycle
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-Flaass, A. V. Kostochka, E. Sopena, J Graph Theory 40 (2002), 83–90). This conjecture if proved would imply both Borodin's (Discrete Math 25 (1979), 211–236) acyclic 5-color theorem and Thomassen's (J Combin Theory Ser B 62 (1994), 180–181) 5-choosability theorem. However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions are also obtained for a planar graph to be acyclically 4- and 3-choosable. In particular, the acyclic 4-choosability was proved for the following planar graphs: without 3-, 4-, and 5-cycles (M. Montassier, P. Ochem, and A. Raspaud, J Graph Theory 51 (2006), 281–300), without 4-, 5-, and 6-cycles, or without 4-, 5-, and 7-cycles, or without 4-, 5-, and intersecting 3-cycles (M. Montassier, A. Raspaud, W. Wang, Topics Discrete Math (2006), 473–491), and neither 4- and 5-cycles nor 8-cycles having a triangular chord (M. Chen and A. Raspaud, Discrete Math. 310(15–16) (2010), 2113–2118). The purpose of this paper is to strengthen these results by proving that each planar graph without 4- and 5-cycles is acyclically 4-choosable.