Rainbow directed version of Dirac's theorem

Abstract

Let $G=\{G_i:i\in [s\left]\right\}$ be a collection of not necessarily distinct graphs on the same vertex set V. A graph H is called rainbow in $G$ if any two edges of H belong to different graphs of $G$. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem. In this paper, we prove a rainbow directed version of Dirac's theorem asymptotically: For each $0<\epsilon <1$, there exists an integer N such that when $n\ge N$ the following holds. Let $D=\{D_i:i\in [n\left]\right\}$ be a collection of n-vertex digraphs on the same vertex set V. If both the out-degree and the in-degree of v are at least $(\frac{1}{2}+\epsilon )n$ for each vertex $v\in V$ and each integer $i\in \left[n\right]$, then $D$ contains a rainbow Hamiltonian cycle. Furthermore, we provide a sufficient condition for the existence of arbitrary rainbow tournaments in a collection of n-vertex digraphs, where a tournament is an oriented graph of a complete graph.