Large Sets of $\lambda$-Fold $K_{1,3}$-Factors of Complete Bipartite Graphs

Abstract

Let $G$ be a finite graph and $H$ be a subgraph of $G$. If $V(H)=V(G)$ then the subgraph is called a spanning subgraph of $G$. A spanning subgraph $H$ of $G$ is called an $F$-factor if each component of $H$ is isomorphic to $F$. Further, if there exists a subgraph of $G$ whose vertex set is $V(G)$ and can be partitioned into $F$-factors, then it is called a $\lambda$-fold $F$-factor of $G$, denoted by $S_{\lambda }(1,F,G)$. A large set of $\lambda$-fold $F$-factors of $G$, denoted by $L{S}_{\lambda }(1,F,G)$, is a partition $\{{\mathcal{B}}_i{\}}_i$ of all subgraphs of $G$ isomorphic to $F$, such that each $(X,{\mathcal{B}}_i)$ forms a $\lambda$-fold $F$-factor of $G$. In this paper, we investigate $L{S}_{\lambda }(1,K_{1,3},K_{v,v})$ for any index $\lambda$ and obtain existence results for the cases $v=4t,2t+1,12t+6$ and $v\ge 3$.