A completeness criterion for the common divisor graph on $p$-regular class sizes
Víctor Sotomayor
TL;DR
The paper addresses when the p-regular common divisor graph Γ_p(G) of a finite group G is k-regular, showing that this occurs precisely when Γ_p(G) is a complete graph with k+1 vertices. It develops a sequence of lemmas—including connectedness under regularity and the impact of prime-power class sizes—and then conducts a three-step contradiction argument (extending BCHP/RA methods) to prove the main criterion without assuming p-solubility. Notably, it recovers the classical Γ(G) result when p does not divide |G|, and it demonstrates that Γ_p(G) can be regular even when the ordinary graph is not. The work thus provides a complete completeness criterion for Γ_p(G) and clarifies the possible arithmetical structures of p-regular class sizes across all finite groups.
Abstract
Let $G$ be a finite group. For some fixed prime $p$, let $Γ_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of those non-central elements of $G$ such that $p$ does not divide their order, and two distinct vertices are adjacent if and only if they are not coprime. In this note we prove that if $Γ_p(G)$ is a $k$-regular graph with $k\geq 1$, then it is a complete graph with $k+1$ vertices.