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Decomposition of Sparse Graphs into Forests and a Graph with Bounded Degree

Authors


  • Contract grant sponsor: NSF (to A. V. K.); Contract grant number: DMS-0965587; Contract grant sponsor: NSA (to D. B. W.); Contract grant number: H98230-10-1-0363; Contract grant sponsor: NSFC (to X. Z.); Contract grant number: 11171730; Contract grant sponsor: ZJNSF (to X. Z.); Contract grant number: Z6110786.

Abstract

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

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

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