A degree sequence Koml\'{o}s theorem

An important result of Koml'os [Tiling Tur'an theorems, Combinatorica, 2000] yields the asymptotically exact minimum degree threshold that ensures a graph $G$ contains an $H$-tiling covering an $x$th proportion of the vertices of $G$ (for any fixed $x\in (0,1)$ and graph $H$). We give a degree sequence strengthening of this result which allows for a large proportion of the vertices in the host graph $G$ to have degree substantially smaller than that required by Koml'os’ theorem. We also demonstrate that for certain graphs $H$, the degree sequence condition is essentially best possible in more than one sense.