Disjoint cycles with length constraints in digraphs of large connectivity or minimum degree

TL;DR

This paper investigates conditions under which digraphs contain multiple disjoint directed cycles of different lengths, proving results for high connectivity and minimum degree, and highlighting open problems related to degree thresholds.

Contribution

It establishes that high connectivity guarantees disjoint cycles of distinct lengths and constructs examples showing limitations, advancing understanding of cycle length constraints in digraphs.

Findings

High connectivity ensures disjoint cycles of distinct lengths.

Existence of a degree threshold for three disjoint cycles of different lengths.

Counterexamples show limitations of connectivity in cycle length diversity.

Abstract

A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. In contrast, for every $k \in \mathbb{N}$ we construct a strongly $k$-connected digraph containing no two vertex- or arc-disjoint directed cycles of the same length. It is an open problem whether $g(3)$ exists. Here we prove the existence of an integer $K$ such that every digraph of minimum out- and in-degree at least $K$ contains $3$ vertex-disjoint directed cycles of pairwise distinct lengths.