  A derangement $k$-representation of a graph $G$ is a map $\pi$ of $V(G)$ to
the symmetric group $S_k$, such that for any two vertices $v$ and $u$ of
$V(G)$, $v $ and $u$ are adjacent if and only if $\pi(v)(i) \neq \pi(u)(i)$ for
each $i \in \{1,2,3,\ldots,k\}$. The derangement representation number of $G$
denoted by $drn(G)$, is the minimum of $k$ such that $G$ has a derangement
$k$-representation. In this paper, we prove that any graph has a derangement
$k$-representation. Also, we obtain some lower and upper bounds for $drn(G)$,
in terms of the basic parameters of $G$. Finally, we determine the exact value
or give the better bounds of the derangement representation number of some
classes of graphs.
