Packing spanning arborescences with extra large one

TL;DR

This work provides a digraphic analogue of Fang and Yang's result on packing spanning trees with an extra large part. It proves that if $ν_{f}(D) > k+ \frac{d-1}{d}$ for a digraph $D$, then $D$ contains arc-disjoint $k$ spanning arborescences and an additional branching with more than $\frac{d-1}{d}(|V(D)|-1)$ arcs, with the extra branching either spanning or containing a component with at least $d$ arcs. The proof proceeds by induction on the number of vertices, leveraging a key packing lemma for branchings and the properly intersecting elimination operation PIEO$^{1}$, along with contraction arguments. The results deepen the fractional packing theory for digraphs and highlight the open challenge of achieving a corresponding mixed-graphic generalization.

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

Theorems & Definitions (22)

• Theorem 1: N-64
• Theorem 2: N-61T-61
• Conjecture 1: NDT Conjecture, MMRZ-12
• Theorem 3: NDT Theorem, JY-17
• Conjecture 2: GY-23
• Theorem 4: GY-23
• Theorem 5: FY-24
• Theorem 6
• Theorem 7: F-09
• Lemma 1: GY-21