On packing dijoins in digraphs and weighted digraphs

TL;DR

This paper investigates how to partition the arcs of a directed graph into dijoins, especially under weighted conditions, providing new results that extend and refine Woodall's conjecture and related packing problems.

Contribution

The paper proves new theorems on packing dijoins in digraphs with weighted arcs, extending previous conjectures and establishing optimal conditions for such packings.

Findings

Partitioning arcs into a dijoin and a (τ-1)-dijoin is always possible.

Under certain weight conditions, equitable packings of τ dijoins exist.

Specific cases confirm the optimality of the conditions for packings.

Abstract

Let $D=(V,A)$ be a digraph. A dicut is a cut $\delta^+(U)\subseteq A$ for some nonempty proper vertex subset $U$ such that $\delta^-(U)=\emptyset$, a dijoin is an arc subset that intersects every dicut at least once, and more generally a $k$-dijoin is an arc subset that intersects every dicut at least $k$ times. Our first result is that $A$ can be partitioned into a dijoin and a $(\tau-1)$-dijoin where $\tau$ denotes the smallest size of a dicut. Woodall conjectured the stronger statement that $A$ can be partitioned into $\tau$ dijoins. Let $w\in \mathbb{Z}^A_{\geq 0}$ and suppose every dicut has weight at least $\tau$, for some integer $\tau\geq 2$. Let $\rho(\tau,D,w):=\frac{1}{\tau}\sum_{v\in V} m_v$, where each $m_v$ is the integer in $\{0,1,\ldots,\tau-1\}$ equal to $w(\delta^+(v))-w(\delta^-(v))$ mod $\tau$. We prove the following results: (i) If $\rho(\tau,D,w)\in \{0,1\}$, then there is an equitable $w$-weighted packing of dijoins of size $\tau$. (ii) If $\rho(\tau,D,w)= 2$, then there is a $w$-weighted packing of dijoins of size $\tau$. (iii) If $\rho(\tau,D,w)=3$, $\tau=3$, and $w={\bf 1}$, then $A$ can be partitioned into three dijoins. Each result is best possible: (i) does not hold for $\rho(\tau,D,w)=2$ even if $w=\1$, (ii) does not hold for $\rho(\tau,D,w)=3$, and (iii) do not hold for general $w$.