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The $k$th power of a graph $G$, denoted $G^k$, has the same vertex set as$G$, and two vertices are adjacent in $G^k$ if and only if there exists a pathbetween them in $G$ of length at most $k$. A $K_r$-factor in a graph is aspanning subgraph in which every component is a complete graph of order $r$. Itis easy to show that for any connected graph $G$ of order divisible by $r$,$G^{2r-2}$ contains a $K_r$-factor. This is best possible as there existconnected graphs $G$ of order divisible by $r$ such that $G^{2r-3}$ does notcontain a $K_r$-factor. We conjecture that for any 2-connected graph $G$ oforder divisible by $r$, $G^r$ contains a $K_r$-factor. This was known for $r\le 3$ and we prove it for $r = 4$. We prove a stronger statement that thevertex set of any 2-connected graph $G$ of order $4k$ can be partitioned into$k$ parts of size $4$, such that the four vertices in any part are contained ina subtree of $G$ of order at most 5. More generally, we conjecture that for anypartition of $n = n_1+n_2+\cdots+n_k$, the vertex set of any 2-connected graph$G$ of order $n$ can be partitioned into $k$ parts $V_1,V_2,\ldots,V_k$, suchthat $|V_i| = n_i$ and $V_i \subseteq V(T_i)$ for some subtree $T_i$ of $G$ oforder at most $n_i+1$, for $1 \le i \le k$.

Clique factors in powers of graphs