Decomposition of sparse graphs into forests: The Nine Dragon Tree Conjecture for k  2

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Abstract

For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of |E(H)||V(H)|1 over all subgraphs H with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if Arb(G)k+dk+d+1, then G decomposes into k+1 forests with one having maximum degree at most d. The conjecture was previously proved for d=k+1 and for k=1 when d6. We prove it for all d when k2, except for (k,d)=(2,1).

Keywords

Nine Dragon Tree Conjecture
Arboricity
Nash-Williams Arboricity Formula
Fractional arboricity
Forest
Graph decomposition
Discharging method
Sparse graph

Cited by (0)

1

Research supported by NSFC grant 11471293 and ZJNSFC grant LY14A010014.

2

Research supported by Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, grant NRF-2015R1D1A1A01057008.

3

Research supported in part by NSF grants DMS-1266016 and DMS-1600592.

4

Research supported by Recruitment Program of Foreign Experts, 1000 Talent Plan, State Administration of Foreign Experts Affairs, China.

5

Research supported by CNSF grant 11571319.