From finding a spanning subgraph $H$ to an $H$-factor

Abstract

A typical Dirac-type problem in extremal graph theory is to determine the minimum degree threshold for a graph $G$ to have a spanning subgraph $H$, e.g. the Dirac theorem. A natural follow-up problem is to seek an $H$-factor, which is a spanning set of vertex-disjoint copies of $H$. In this short note, we present a method for obtaining an upper bound on the minimum degree threshold for an $H$-factor from one for finding a spanning copy of $H$. As an application, we prove that, for all $\epsilon >0$ and sufficiently large $\ell$, any oriented graph $G$ on $\ell m$ vertices with minimum semi-degree ${\delta }^0(G)\ge (3/8+\epsilon )\ell m$ contains a $C_{\ell }$-factor, where $C_{\ell }$ is an arbitrary orientation of a cycle on $\ell$ vertices. This improves a result of Wang, Yan and Zhang.

arXiv commentminor revision, to be published at Journal of Graph Theory