n-exact Character Graphs

Let $\Gamma$ be a finite simple graph. If for some integer $n\geqslant 4$, $\Gamma$ is a $K_n$-free graph whose complement has an odd cycle of length at least $2n-5$, then we say that $\Gamma$ is an $n$-exact graph. For a finite group $G$, let $\Delta(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. 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 $\Delta(G)$.
Comment: arXiv admin note: substantial text overlap with arXiv:1909.01180, arXiv:1909.03062, arXiv:1907.13292