Triangle-tilings in graphs without large independent sets

We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an $n$-vertex graph $G$ with sublinear independence number. In this setting, we show that if $\delta(G) \ge n/3 + o(n)$ then $G$ has a triangle-tiling covering all but at most four vertices. Also, for every $r \ge 5$, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that $G$ is $K_r$-free and $n$ is divisible by $3$.
Comment: 24 pages