Further progress on Wojda's conjecture

TL;DR

This paper advances the understanding of Wojda's conjecture by confirming it for larger values of m and n, demonstrating that certain size conditions guarantee the packing of two digraphs.

Contribution

The paper proves Wojda's conjecture for all m ≥ 93 and n ≥ 31m, extending previous results to a broader range of parameters.

Findings

Confirmed Wojda's conjecture for m ≥ 93 and n ≥ 31m

Established new bounds for digraph packing conditions

Extended the range of parameters where the conjecture holds

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 $m = \Omega(\sqrt{n})$. In this paper, we confirm it for $m \geq 93$ and $n \geq 31m$.