An approximate version of Jackson's conjecture

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 $\epsilon >0$ there exists $n_0$ such that every $cn$-regular bipartite digraph on $2n\ge n_0$ vertices contains $(1-\epsilon )cn$ edge-disjoint Hamilton cycles.