Packing spanning arborescences with extra large one

Abstract

The celebrated Nash-Williams and Tutte's theorem states that a graph $G=(V,E)$ contains $k$ edge disjoint spanning trees if and only if $\hat{I}{½}_f(G)\ge k$, where $$\hat{I}{½}_f(G):=\underset{|\mathcal{P}|>1,\text{P is a partition of V(G)}}{\min }\frac{|E(\mathcal{P})|}{|\mathcal{P}|-1}.$$ Inspired by the NDT theorem as structural explanations for the fractional part of Nash-Williams' forest decomposition theorem, Fang and Yang extended Nash-Williams and Tutte's theorem and proved that if $\hat{I}{½}_f(G)>k+\frac{d-1}{d}$, then $G$ contains $k$ edge disjoint spanning trees and another forest $F$ with $|E(F)|>\frac{d-1}{d}(|V(G)|-1)|$, and if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. In this paper, we give a digraphic version of their result; however, the mixed graphic version remains open.