On the Codegree graphs of finite groups

The codegree of an irreducible character $\chi$ of a finite group $G$ is defined as $\mathrm{\mid }G:\ker \chi \mathrm{\mid }\mathrm{/}\chi (1)$. The codegree graph $\mathrm{\Gamma }(G)$ of a finite group $G$ is the graph whose vertices are the prime divisors of $\mathrm{\mid }G\mathrm{\mid }$, where two distinct primes $p$ and $q$ are adjacent if and only if $pq$ divides the codegree of some irreducible character of $G$. In this paper, we prove that a graph can occur as a codegree graph $\mathrm{\Gamma }(G)$ of some finite group $G$ if and only if its complement is triangle-free and $3$-colorable. This generalizes the known characterization for codegree graphs from solvable groups to all finite groups. As an application, we give a full classification of all groups for which $\mathrm{\Gamma }(G)$ is a $5$-cycle. We also investigate conditions under which the codegree graph coincides with or differs from the prime graph for solvable groups.