A Degree Condition for the Existence of Regular Factors in K₁, _n-Free Graphs

A graph is called K₁, _n-free if it contains no K₁, _n as an induced subgraph. Let n(≥ 3), r be integers (if r is odd, r ≥ n - 1). We prove that every K₁, _n-free connected graph G with r|V(G)| even has an r-factor if its minimum degree is at least (n + n-1/r) ⌈n/2(n - 1) r⌉ - n - 1/r (⌈n/2(n - 1) r⌉)² + n - 3. This degree condition is sharp.