$F$-Factor and Vertex-Disjoint $F$ in a Graph

Abstract

Let $G$ be a graph of order $n\ge 4k$ and let $S$ be the graph obtained from $K_4$ by removing two edges which have a common vertex. In this paper, we prove the following theorem:
A graph $G$ of order $n\ge 4k$ with ${\sigma }_2(G)\ge n+k$ has $k$ vertex-disjoint $S$.This theorem implies that a graph $G$ of order $n=4k$ with ${\sigma }_2(G)\ge 5k$ has an $S$-factor.