Complementary cycles of any length in regular bipartite tournaments

TL;DR

This paper proves that in regular bipartite tournaments, for most cycle lengths, there exist complementary cycles such that removing them leaves a Hamiltonian digraph, confirming a longstanding conjecture.

Contribution

It establishes the existence of complementary cycles of various lengths in regular bipartite tournaments, extending previous results and confirming a conjecture.

Findings

Existence of cycles of length 2p for 2 ≤ p ≤ n/2-2

Complementary cycles leave a Hamiltonian digraph unless the tournament is isomorphic to F_{4k}

Confirmed the conjecture for all applicable p values

Abstract

Let $D$ be a $k$-regular bipartite tournament on $n$ vertices. We show that, for every $p$ with $2 \le p \le n/2-2$, $D$ has a cycle $C$ of length $2p$ such that $D \setminus C$ is hamiltonian unless $D$ is isomorphic to the special digraph $F_{4k}$. This statement was conjectured by Manoussakis, Song and Zhang [K. Zhang, Y. Manoussakis, and Z. Song. Complementary cycles containing a fixed arc in diregular bipartite tournaments. Discrete Mathematics, 133(1-3):325--328,1994]. In the same paper, the conjecture was proved for $p=2$ and more recently Bai, Li and He gave a proof for $p=3$ [Y. Bai, H. Li, and W. He. Complementary cycles in regular bipartite tournaments. Discrete Mathematics, 333:14--27, 2014].