Connected [a,b]-factors in K1,n-free graphs containing an [a,b]-factor

A graph G is called K1,n-free if G has no induced subgraph isomorphic to K1,n. Let n, a, and b be integers with n⩾3, a⩾1, and b⩾a(n−2)+2. We prove that every connected K1,n-free graph G has a connected [a,b]-factor if G contains an [a,b]-factor. This result is sharp in the sense that there exists a connected K1,n-free graph which has an [a,b]-factor but no connected [a,b]-factor for b⩽a(n−2)+1.