We prove that for $r\in \mathbb{ N}$ with $r\geq 2$ and $\mu>0$, there exist $\alpha>0$ and $n_{ 0}$ such that for every $n\geq n_{ 0}$, every $n$-vertex graph $G$ with $\delta(G)\geq \left(1-\frac{ 1}{ r}+\mu\right)n$ and $\alpha(G)\leq \alpha n$ contains an $r$-th power of a Hamilton cycle. We also show that the minimum degree condition is asymptotically sharp for $r=2, 3$ and the $r=2$ case was recently conjectured by Staden and Treglown.
Categories: math.CO