$K_n$-free Character Graphs with at Least $2n$ Vertices

For a finite group $G$, let $\mathrm{\Delta }(G)$ 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 $n$, $\mathrm{\Delta }(G)$ is $K_n$-free, then $\mathrm{\Delta }(G)$ 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.