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 \(LS_\lambda(1,F,G)\), 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 \(LS_\lambda(1,K_{1,3},K_{v,v})\) for any index \(\lambda\) and obtain existence results for the cases \(v = 4t, 2t + 1, 12t+6\) and \(v \geq 3\).