# On the character degree graph of solvable groups

## TL;DR

This paper extends a known result about the prime graph of solvable groups, showing its complement is bipartite, which leads to bounds on the graph's size and supports broader conjectures in group theory.

## Contribution

It generalizes Pálfy's theorem, proving the complement of the prime graph is bipartite, and confirms a conjecture relating to the structure of these graphs in solvable groups.

## Key findings

- The complement of the prime graph contains no odd cycles.
- The vertices of the prime graph can be covered by two complete subgraphs.
- The prime graph has at most twice as many vertices as its clique number.

## Abstract

Let \(G\) be a finite solvable group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). A fundamental result by P.P. P\'alfy asserts that the complement $\bar{\Delta}(G)$ of the graph \(\Delta(G)\) does not contain any cycle of length \(3\). In this paper we generalize P\'alfy's result, showing that $\bar{\Delta}(G)$ does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of \(\Delta(G)\) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if \(n\) is the clique number of \(\Delta(G)\), then \(\Delta(G)\) has at most \(2n\) vertices. This confirms a conjecture by Z. Akhlaghi and H.P. Tong-Viet, and provides some evidence for the famous \emph{\(\rho\)-\(\sigma\) conjecture} by B. Huppert.