A relaxation of the Bermond-Thomassen conjecture

2026-08-13

AI Summary

This study addresses the existence of cycles in the context of the Bermond–Thomassen conjecture for $k \geq 4$. The conjecture posits that every directed graph with minimum out-degree at least $2k - 1$ contains $k$ vertex-disjoint directed cycles. We propose and prove a tight relaxation of this conjecture, establishing that such graphs necessarily contain $k$ vertex-disjoint cycles, each of which is either directed or can be made directed by reversing a single arc. This result resolves an open problem posed by Cames van Batenburg (2021). By integrating structural graph theory with extremal combinatorial methods, we develop a new cycle existence theorem whose bound is best possible, thereby significantly advancing the understanding of this classical conjecture.

Abstract

The well-known Bermond-Thomassen conjecture states that every digraph of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint directed cycles. Despite being posed in 1981, this conjecture remains unresolved for all $k \ge 4$. We prove a relaxation of this conjecture: every digraph $D$ of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint cycles, each of which either is directed or can be made directed by reversing one of its arcs. This bound is sharp and answers a question raised by Cames van Batenburg during the online workshop "Entropy Compression and Related Methods" in $2021$.

Problem

• Bermond-Thomassen conjecture
• directed cycles
• vertex-disjoint cycles
• minimum out-degree
• digraph

Innovation

• Bermond-Thomassen conjecture
• vertex-disjoint cycles
• directed graphs
• arc reversal
• minimum out-degree