# Partition of a Subset into Two Directed Cycles with Partial Degrees

## TL;DR

This paper proves that under certain degree conditions, a directed graph contains two disjoint cycles covering a specified subset, and conjectures a similar property for multiple cycles with a partition of the subset.

## Contribution

It establishes a degree condition ensuring two disjoint cycles cover a subset with a given partition, and proposes a conjecture for multiple cycles with sharper conditions.

## Key findings

- Degree condition guarantees two disjoint cycles covering a subset
- Conjecture extends the result to multiple cycles with larger partitions
- Degree condition is shown to be sharp in general

## Abstract

Let $D=(V,A)$ be a directed graph of order $n\geq 6$. Let $W$ be a subset of $V$ with $|W|\geq 6$. Suppose that every vertex of $W$ has degree at least $(3n-3)/2$ in $D$. Then for any integer partition $|W|=n_1+n_2$ with $n_1\geq 3$ and $n_2\geq 3$, $D$ contains two disjoint directed cycles $C_1$ and $C_2$ such that $|V(C_1)\cap W|=n_1$ and $|V(C_2)\cap W|=n_2$. We conjecture that for any integer partition $|W|=n_1+n_2+\cdots +n_k$ with $k\geq 3$ and $n_i\geq 3(1\leq i\leq k)$, $D$ contains $k$ disjoint directed cycles $C_1,C_2,\ldots , C_k$ such that $|V(C_i)\cap W|=n_i$ for all $1\leq i\leq k$. The degree condition is sharp in general.