# An approximate version of Jackson's conjecture

## TL;DR

This paper proves an approximate version of Jackson's conjecture, demonstrating that dense regular bipartite digraphs contain nearly a full set of edge-disjoint Hamilton cycles, extending understanding of Hamiltonian decompositions.

## Contribution

It establishes that for sufficiently large dense bipartite digraphs, a near-complete Hamilton cycle decomposition exists, advancing the theory of Hamiltonian decompositions in directed graphs.

## Key findings

- Dense bipartite digraphs contain almost all edges in Hamilton cycles.
- The result applies to sufficiently large regular bipartite digraphs.
- Nearly all edges can be covered by edge-disjoint Hamilton cycles.

## Abstract

In 1981 Jackson showed that the diregular bipartite tournament (a complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree) contains a Hamilton cycle, and conjectured that in fact the edge set of it can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: For every $c>1/2$ and $\varepsilon>0$ there exists $n_0$ such that every $cn$-regular bipartite digraph on $2n\geq n_0$ vertices contains $(1-\varepsilon)cn$ edge-disjoint Hamilton cycles.