  The scaling behavior of self-avoiding walks (SAWs) on the backbone of
percolation clusters in two, three and four dimensions is studied by Monte
Carlo simulations. We apply the pruned-enriched Rosenbluth chain-growth method
(PERM). Our numerical results bring about the estimates of critical exponents,
governing the scaling laws of disorder averages of the end-to-end distance of
SAW configurations. The effects of finite-size scaling are discussed as well.
