Digraph analogues for the Nine Dragon Tree Conjecture

The fractional arboricity of a digraph D $D$ , denoted by γ ( ) $\gamma (D)$ is defined as = max H ⊆ | V > 1 A − (D)=\mathop{\max }\limits_{H\subseteq D,|V(H)|\gt 1}\frac{|A(H)|}{|V(H)|-1}$ . Frank proved that decomposes into k $k$ branchings, if and only Δ ≤ ${{\rm{\Delta }}}^{-}(D)\le k$ (D)\le In this paper, we study analogues for the Nine Dragon Tree Conjecture. We conjecture that, positive integers d $d$ with + k+\frac{d-k}{d+1}$ k+1$ then $k+1$ branchings B … ${B}_{1},\ldots ,{B}_{k},{B}_{k+1}$ }}}^{+}({B}_{k+1})\le d$ This conjecture, true, refinement Frank's characterization. series acyclic bipartite digraphs also presented to show bound given in best possible. prove our cases $d\le As more evidence support maximum average degree mad 2 $\,\text{mad}\,(D)\le 2k+\frac{2(d-k)}{d+1}$ pseudo-branchings C ${C}_{1},\ldots ,{C}_{k},{C}_{k+1}$ }}}^{+}({C}_{k+1})\le