Large Sets of $\lambda$-Fold $P_3$-Factors in $K_{v,v}$

Abstract

Let $G$ be a finite graph and $H$ be a subgraph of $G$. If $V(H)=V(G)$, then the subgraph $H$ is called a $spanningsubgraph$ 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 $\lambda V(G)$ and can be partitioned into $F$-factors, then it is called a $\lambda -foldF-factor$ of $G$, denoted by $S_{\lambda }(1,F,G)$. A $largeset$ of $\lambda$-fold $F$-factors in $G$ 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 the large set of $\lambda$-fold $P_3$-factors in $K_{v,v}$ and obtain its existence spectrum.