This work was supported by Konkuk University.
Decomposition of Sparse Graphs into Forests and a Graph with Bounded Degree
Article first published online: 3 JAN 2013
© 2013 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 74, Issue 4, pages 369–391, December 2013
How to Cite
Kim, S.-J., Kostochka, A. V., West, D. B., Wu, H. and Zhu, X. (2013), Decomposition of Sparse Graphs into Forests and a Graph with Bounded Degree. J. Graph Theory, 74: 369–391. doi: 10.1002/jgt.21711
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.
- Issue published online: 28 OCT 2013
- Article first published online: 3 JAN 2013
- Manuscript Revised: 15 NOV 2012
- Manuscript Received: 12 APR 2012
- Konkuk University
- NSF. Grant Number: DMS-0965587
- NSA. Grant Number: H98230-10-1-0363
- NSFC. Grant Number: 11171730
- ZJNSF. Grant Number: Z6110786
- graph decomposition;
- fractional arboricity;
- maximum average degree;
For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of over all subgraphs H with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if , then G decomposes into forests with one having maximum degree at most d. The conjecture was previously proved for ; we prove it for and when and . For , we can further restrict one forest to have at most two edges in each component.
For general , we prove weaker conclusions. If , then implies that G decomposes into k forests plus a multigraph (not necessarily a forest) with maximum degree at most d. If , then implies that G decomposes into forests, one having maximum degree at most d. Our results generalize earlier results about decomposition of sparse planar graphs.