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

Abstract

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.

arXiv admin note: text overlap with arXiv:2002.01353. The counterexample given in this manuscripts has been presented in [Bounding the number of vertices in the degree graph of a finite group, Z. Akhlaghi, S. Dolfi, E. Pacifici, L. Sanus, Journal of Pure and Applied Algebra, 224(2020) 725-731

References in corpus (1)

• Groups whose prime graphs have no triangles