n-exact Character Graphs

Let $Γ$ be a finite simple graph. If for some integer $n\geqslant 4$, $Γ$ is a $K_n$-free graph whose complement has an odd cycle of length at least $2n-5$, then we say that $Γ$ is an $n$-exact graph. For a finite group $G$, let $Δ(G)$ 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 $Δ(G)$.

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