The Overfull Nine Dragon Tree Conjecture is True

Chen, Kim, Kostochka, West, and Zhu conjectured a strengthening of the Nine Dragon Tree Theorem that every graph that is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -sparse and has no overfull set decomposes into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> forests such that one of the forests has maximum degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> . Intuitively, the conjecture says that while the fractional arboricity bound in the Nine Dragon Tree Theorem is tight, we can relax the bound very slightly and still get the same result, so long as the obvious obstruction does not occur - that we can still decompose into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> forests. We prove this conjecture.