Further progress on Wojda's conjecture

TL;DR

The paper tackles Wojda's conjecture on packing two digraphs in the complete digraph by focusing on the regime where one digraph is near size $n$ and the other is of size $n-m$. It develops probabilistic near-packing tools, degree-based bounds, and a decomposition into oriented trees to extend packings from small components to full graphs. The main result verifies the conjecture for all $m \ge 93$ and $n \ge 31m$, showing $\mu(n,n-m)=2n-\left\lfloor \frac{n}{m} \right\rfloor$. This sharpens the extremal function for digraph packing in a substantial parameter range and advances understanding of when edge-disjoint packings exist in complete digraphs.

Abstract

Two digraphs of order $n$ are said to pack if they can be found as edge-disjoint subgraphs of the complete digraph of order $n$. It is well established that if the sum of the sizes of the two digraphs is at most $2n-2$, then they pack, with this bound being sharp. However, it is sufficient for the size of the smaller digraph to be only slightly below $n$ for the sum of their sizes to significantly exceed this threshold while still guaranteeing the existence of a packing. In 1985, Wojda conjectured that for any $2 \leq m \leq n/2$, if one digraph has size at most $n - m$ and the other has size less than $2n - \lfloor n/m \rfloor$, then the two digraphs pack. It was previously known that this conjecture holds for $m = Ω(\sqrt{n})$. In this paper, we confirm it for $m \geq 93$ and $n \geq 31m$.

Figures (1)

• Figure 1: Digraphs $D$ and $D'$ do not pack

Theorems & Definitions (10)

• Definition 1
• Conjecture 2
• Theorem 3
• Theorem 4
• Definition 5
• Lemma 6
• Theorem 7
• Lemma 8
• Proposition 9
• Claim 10