The t-improper chromatic number of random graphs

For a non-hamiltonian claw-free graph G with order n and minimum degree δ we show the following. If δ = 4 , then G has a 2-factor with at most ( 5 n − 14 ) / 18 components, unless G belongs to a finite class of exceptional graphs. If δ ⩾ 5 , then G has a 2-factor with at most ( n − 3 ) / ( δ − 1 ) components. These bounds are sharp in the sense that we can replace nor 5/18 by a smaller quotient nor δ − 1 by δ.