We consider a kind of Cayley graphs, including the modified bubble-sort graphs MBn the Cayley graphs generated by lollipop graphs H-n,H-n-1 and the other Cayley graphs generated by unicyclic triangle free graphs. We get that if we delete linearly many vertices in Cayley graphs generated by transposition unicyclic triangle free graphs, then the resulting graphs has a large connected component that comprises of nearly all residual vertices. (C) 2020 Elsevier B.V. All rights reserved.