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

Ken-ichi Kawarabayashi, Carsten Thomassen

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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 Grötzsch'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 Grötzsch's theorem an unbounded number of triangles are allowed.
    Original languageEnglish
    JournalJournal of Combinatorial Theory. Series B
    Volume99
    Issue number4
    Pages (from-to)674-684
    ISSN0095-8956
    DOIs
    Publication statusPublished - 2009

    Keywords

    • Forests
    • Independent sets
    • Planar graphs of girth 5

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