n-exact Character Graphs

Let $\mathrm{\Gamma }$ be a finite simple graph. If for some integer $n⩾4$, $\mathrm{\Gamma }$ is a $K_n$-free graph whose complement has an odd cycle of length at least $2n-5$, then we say that $\mathrm{\Gamma }$ is an $n$-exact graph. 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 $G$. In this paper, we prove that the order of an $n$-exact character graph is at most $2n-1$. Also we determine the structure of all finite groups $G$ with extremal $n$-exact character graph $\mathrm{\Delta }(G)$.