Partition of a Subset into Two Directed Cycles with Partial Degrees

Abstract

Let $D=(V,A)$ be a directed graph of order $n\ge 6$. Let $W$ be a subset of $V$ with $|W|\ge 6$. 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 $n_1\ge 3$ and $n_2\ge 3$, $D$ 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 $k\ge 3$ and $n_i\ge 3(1\le i\le k)$, $D$ contains $k$ disjoint directed cycles $C_1,C_2,\dots ,C_k$ such that $|V(C_i)\cap W|=n_i$ for all $1\le i\le k$. The degree condition is sharp in general.