Decomposing a planar graph of girth 5 into an independent set and a forest

  • Authors:

  • Ken-ichi Kawarabayashi;Carsten Thomassen

  • Affiliations:

  • National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan;Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2009

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Abstract

We use a list-color technique to extend the result of Borodin and Glebov that the vertex set of every planar graph of girth at least 5 can be partitioned into an independent set and a set which induces a forest. We apply this extension to also extend Grotzsch's theorem that every planar triangle-free graph is 3-colorable. Let G be a plane graph. Assume that the distance between any two triangles is at least 4. Assume also that each triangle contains a vertex such that this vertex is on the outer face boundary and is not contained in any 4-cycle. Then G has chromatic number at most 3. Note that, in this extension of Grotzsch's theorem an unbounded number of triangles are allowed.