Minimum degree conjecture for perfect clique tilings in multipartite graphs

Let $k\geq r\geq 2$ and $\gamma>0$. Let $G$ be a $k$-partite graph on $n$ vertices with parts $V_1,\ldots,V_k$ such that $|V_i|\leq n/r$ for every $i$. For each $i\in [k]$, write $\delta(V_i)$ for the minimum degree of a vertex in $V_i$. Multipartite perfect $K_r$-tiling conjecture. If

$$\delta(V_i)\geq \left(1-\frac{1}{2r}+\gamma\right)n-|V_i|$$

for all $i\in [k]$, then $G$ has a perfect $K_r$-tiling.

This conjecture proposes a sufficient minimum total-degree condition for perfect $K_r$-tilings in unbalanced multipartite graphs and is asymptotically necessary in certain cases. It is presented as an open direction beyond known balanced results, including the asymptotically best possible balanced multipartite threshold proved by Lo and Sanhueza-Matamala.