  For a finite group $G$, let $\Delta(G)$ denote the character graph built on
the set of degrees of the irreducible complex characters of $G$. Akhlaghi and
Tong-Viet in \cite{[AT]} conjectured that if for some positive integer $n$,
$\Delta(G)$ is $K_n$-free, then $\Delta(G)$ has at most $2n-1$ vertices. In
this paper, we present an example to show that this conjecture is not
necessarily true for all non-solvable groups whose character graphs are
$K_n$-free.
