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We prove the Strong Nine Dragon Tree Conjecture is true if we replace d with d + k2 (d{k+1} - 1) d{k+1}. More precisely: let G be a graph and let d and k be positive integers. If (G) k + dd + k + 1, then there is a partition into k + 1 forests, where in one forest every connected component has at most d + k2 (d{k+1} - 1) d{k+1} edges.
Mies et al. (Fri,) studied this question.
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