Tiling tripartite graphs with 3-colorable graphs: The extreme case

There is a sufficiently large $N\in h\mathbb{N}$ such that the following holds. If $G$ is a tripartite graph with $N$ vertices in each vertex class such that every vertex is adjacent to at least $2N\mathrm{/}3+2h-1$ vertices in each of the other classes, then $G$ can be tiled perfectly by copies of $K_{h,h,h}$. This extends work by two of the authors [Electron. J. Combin, 16(1), 2009] and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that $2N\mathrm{/}3+2h-1$ in our result can not be replaced by $2N\mathrm{/}3+h-2$ and that if $N$ is divisible by $6h$, then we can replace it with the value $2N\mathrm{/}3+h-1$ and this is tight.