Vertex-Disjoint Copies of K1+(K1∪K2) in Graphs

Abstract

Let $S$ denote the graph obtained from $K_4$ by removing two edges which have an endvertex in common. Let $k$ be an integer with $k\ge 2$. Let $G$ be a graph with $|V(G)|\ge 4k$ and ${\sigma }_2(G)\ge |V(G)|/2+2k-1$, and suppose that $G$ contains $k$ vertex-disjoint triangles. In the case where $|V(G)|=4k+2$, suppose further that $G\cancel{\cong }{K}_{4t+3}{{\displaystyle \cup }}^{\text{}}{K}_{4k-4t-1}$ for any $t$ with $0\le t\le k-1$. Under these assumptions, we show that $G$ contains $k$ vertex-disjoint copies of $S$.