Before revision:  For bounded degree minor-free graphs we introduce the notion of  covering partition oracles which is a relaxed version of partition oracles and design a poly(d/\epsilon)-time covering partition oracle for this family of graphs . Using our covering partition oracle we provide the same results as above (except that the tester for Hamiltonicity has one sided error)
After revision:  For bounded degree minor-free graphs we introduce the notion of  covering partition oracles which is a relaxed version of partition oracles and design a poly(d/\epsilon)-time covering partition oracle . Using our covering partition oracle we provide the same results as above (except that the tester for Hamiltonicity has one sided error)