Two disjoint cycles in digraphs

TL;DR

This paper investigates conditions on outdegree sequences in directed graphs that guarantee the existence of multiple disjoint cycles, confirming the conjecture for small values of k and characterizing all such sequences for k ≤ 2.

Contribution

The paper extends Bermond and Thomassen's conjecture by characterizing all outdegree sequences that force k disjoint cycles for k ≤ 2, and verifies the conjecture for these cases.

Findings

Confirmed the conjecture for k ≤ 2.

Provided a complete characterization of outdegree sequences forcing k ≤ 2 disjoint cycles.

Extended understanding of cycle structures in digraphs with specified outdegree conditions.

Abstract

Bermond and Thomassen conjectured that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex disjoint cycles. So far the conjecture was verified for $k\le 3$. Here we generalise the question asking for all outdegree sequences which force $k$ vertex disjoint cycles and give the full answer for $k\le 2$.