On the Codegree graphs of finite groups

TL;DR

The paper resolves the realizability problem for codegree graphs of finite groups by proving that a graph $Γ(G)$ occurs precisely when its complement $\overline{Γ}$ is triangle-free and $3$-colorable, thereby extending solvable-group results to all finite groups. It introduces the Frobenius digraph as a central tool to study edge orientations and relies on Hall-subgroup and solvable-radical arguments to establish both sufficiency and necessity, with $Γ_e(G)\subseteq Γ(G)$ playing a key role. As concrete outcomes, it provides a complete classification of all groups with $Γ(G)$ a $5$-cycle and characterizes when the codegree graph equals the prime graph for solvable groups. These results deepen the understanding of how character-theoretic codegrees reflect and constrain group structure and offer a graph-theoretic framework for recognizing codegree patterns.

Abstract

The codegree of an irreducible character $χ$ of a finite group $G$ is defined as $|G:\kerχ|/χ(1)$. The codegree graph $Γ(G)$ of a finite group $G$ is the graph whose vertices are the prime divisors of $|G|$, 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 $Γ(G)$ 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 $Γ(G)$ is a $5$-cycle. We also investigate conditions under which the codegree graph coincides with or differs from the prime graph for solvable groups.

Figures (3)

• Figure 1: 5-cycle and its Frobenius digraph.
• Figure 2: An example of Frobenius digraph whose prime graph coincides codegree graph
• Figure 3: An arbitrary vertices example of Frobenius digraph whose prime graph coincides codegree graph

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Corollary 2.6
• Definition 3.1