Bipartitions of highly connected tournaments

TL;DR

This paper proves that highly connected tournaments can be partitioned into two parts with each part and their bipartite subgraph still highly connected, advancing understanding of connectivity in directed graphs.

Contribution

It establishes a new result showing that strongly connected tournaments can be partitioned into two strongly connected subgraphs with a strongly connected bipartite subgraph, confirming conjectures for tournaments.

Findings

Existence of such partitions in highly connected tournaments

Quantitative bounds on connectivity needed

Advancement of Thomassen's conjectures for directed graphs

Abstract

We show that if $T$ is a strongly $10^9k^6\log(2k)$-connected tournament, there exists a partition $A, B$ of $V(T)$ such that each of $T[A]$, $T[B]$ and $T[A,B]$ is strongly $k$-connected. This provides tournament analogues of two partition conjectures of Thomassen regarding highly connected graphs.