Georgia Institute of Technology College of Sciences

A diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree. 
In 1981, Jackson showed that a diregular bipartite tournament 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 $\epsilon>0$ there exists $n_0$ such that every diregular bipartite tournament on $2n>n_0$  vertices contains a collection of $(1/2-\epsilon)n$ cycles of length at least $(2-\epsilon)n$. 
Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every $c>1/2$ and $\epsilon>0$ there exists $n_0$ such that every $cn$-regular bipartite digraph on $2n>n_0$ vertices contains $(1-\epsilon)cn$ edge-disjoint Hamilton cycles.

Base on joint work with Yanitsa Pehova, see https://arxiv.org/abs/1907.08479

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