Abstract

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. We prove that if a triangle-free planar graph is not 3-choosable, then it contains a 4-cycle that intersects another 4- or 5-cycle in exactly one edge. This strengthens Thomassen's result [C. Thomassen, J. Combin. Theory Ser. B, 64 (1995), pp. 101–107] that every planar graph of girth at least 5 is 3-choosable. In addition, this implies that every triangle-free planar graph without 6- and 7-cycles is 3-choosable.

MSC codes

  1. 05C15

Keywords

  1. planar graph
  2. triangle-free graph
  3. coloring
  4. list coloring
  5. choosability

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 934 - 945
ISSN (online): 1095-7146

History

Submitted: 8 December 2008
Accepted: 1 June 2010
Published online: 12 August 2010

MSC codes

  1. 05C15

Keywords

  1. planar graph
  2. triangle-free graph
  3. coloring
  4. list coloring
  5. choosability

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