Perfect matchings avoiding prescribed edges in a star-free graph

Aldred and Plummer (1999) have proved that every m -connected K 1 , m - k + 2 -free graph of even order contains a perfect matching which avoids k prescribed edges. They have also proved that the result is best possible in the range 1 ? k ? 1 2 ( m + 1 ) . In this paper, we show that if 1 2 ( m + 2 ) ? k ? m - 1 , their result is not best possible. We prove that if m ? 4 and 1 2 ( m + 2 ) ? k ? m - 1 , every K 1 , ? 2 m - k + 4 3 ? -free graph of even order contains a perfect matching which avoids k prescribed edges. While this is a best possible result in terms of the order of a forbidden star, if 2 m - k + 4 ? 0 ( mod 3 ) , we also prove that only finitely many sharpness examples exist.