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Keywords:

  • graph decomposition;
  • forest;
  • fractional arboricity;
  • maximum average degree;
  • discharging

Abstract

For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of inline image over all subgraphs H with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if inline image, then G decomposes into inline image forests with one having maximum degree at most d. The conjecture was previously proved for inline image; we prove it for inline image and when inline image and inline image. For inline image, we can further restrict one forest to have at most two edges in each component.

For general inline image, we prove weaker conclusions. If inline image, then inline image implies that G decomposes into k forests plus a multigraph (not necessarily a forest) with maximum degree at most d. If inline image, then inline image implies that G decomposes into inline image forests, one having maximum degree at most d. Our results generalize earlier results about decomposition of sparse planar graphs.