Packing spanning arborescences with extra large one

TL;DR

This paper extends classical graph theorems on edge-disjoint spanning trees to directed graphs, providing new conditions for their existence and properties, inspired by fractional and structural graph theory concepts.

Contribution

It offers a digraphic version of Fang and Yang's extension of Nash-Williams and Tutte's theorem, advancing understanding of spanning structures in directed graphs.

Findings

Provides a digraphic analogue of Fang and Yang's theorem

Establishes new conditions for edge-disjoint spanning trees in directed graphs

Highlights open problems in mixed graphic versions

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 $\nu_{f}(G) \geq k$, where $$\nu_{f}(G):=\min_{|\mathcal{\mathcal{P}}|>1, \text{$\mathcal{P}$ is a partition of $V(G)$}}\frac{|E( \mathcal{P})|}{|\mathcal{P}|-1}.$$ 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 $\nu_{f}(G) > k+ \frac{d-1}{d}$, 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.