$K_5^{-}$ -factor in a Graph

Abstract

Let $G$ be a graph and let $\delta (G)$ denote the minimum degree of $G$. Let $F$ be a given connected graph. Suppose that $|V(G)|$ is a multiple of $|V(F)|$. A spanning subgraph of $G$ is called an $F$-factor if its components are all isomorphic to $F$. In 2002, Kawarabayashi [5] conjectured that if $G$ is a graph of order $n$ ($n\ge 3$) with $\delta (G)\ge \frac{{\ell }^2-3\ell +1}{\ell -2}$, then $G$ has a $K_{\ell }^{-}$-factor, where $K_{\ell }^{-}$ is the graph obtained from $K_{\ell }$ by deleting just one edge. In this paper, we prove that this conjecture is true when $\ell =5$.