Hamilton decompositions of regular tripartite tournaments

TL;DR

This paper proves that large regular tripartite tournaments can be approximately decomposed into Hamilton cycles, supporting a conjecture despite known counterexamples, and extends results to balanced tripartite digraphs.

Contribution

It establishes an approximate Hamilton decomposition for regular tripartite tournaments and extends the result to large balanced tripartite digraphs with high degree.

Findings

Regular tripartite tournaments admit approximate Hamilton decompositions.

Large balanced tripartite digraphs with degree close to n have Hamilton decompositions.

Supports the conjecture of K"uhn and Osthus in an approximate sense.

Abstract

K\"uhn and Osthus conjectured in 2013 that regular tripartite tournaments are decomposable into Hamilton cycles. Somewhat surprisingly, Granet gave a simple counterexample to this conjecture almost 10 years later. In this paper, we show that the conjecture of K\"uhn and Osthus nevertheless holds in an approximate sense, by proving that every regular tripartite tournament admits an approximate decomposition into Hamilton cycles. We also study Hamilton cycle packings of directed graphs in the same regime, and show that for large $n$, every balanced tripartite digraph on $3n$ vertices which is $d$-regular for $d\ge (1+o(1))n$ admits a Hamilton decomposition.