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 \({spanning \;subgraph}\) 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 \(\lambda V(G)\) 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 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.