On the Number of Disconnected Character Degree Graphs Satisfying Pálfy's Inequality

TL;DR

The authors study disconnected character degree graphs $\Delta(G)$ of finite solvable groups under Pálfy's condition, showing that such graphs are two cliques whose sizes $a$ and $b$ satisfy $b \ge 2^a-1$. They introduce the graph-order function $c(n)$ to count admissible two-component size-pairs and prove $c(n) = \max\{ \alpha \in Z^+ : n \ge 2^\alpha + \alpha - 1 \}$, yielding exactly the pairs $(n-i,i)$ for $i=1,\dots,c(n)$. They establish that for each $\alpha$ there are exactly $2^\alpha+1$ integers $n$ with $c(n)=\alpha$, with minimal such $n$ at $n=2^\alpha+\alpha+1$ and successive blocks separated by $2^\alpha+1$. Calculations demonstrate that even at enormous sizes, the number of realizable disconnected $\Delta(G)$ remains small (e.g., $c(10^{30})=99$), guiding which graph orders to test for a given $\alpha$ and highlighting the combinatorial sparsity of such graphs.

Abstract

Let $G$ be a finite solvable group with disconnected character degree graph $Δ(G)$. Under these conditions, it follows from a result of Pálfy that $Δ(G)$ consists of two connected components. Another result of Pálfy's gives an inequality relating the sizes of these two connected components. In this paper, we calculate the number of possible component size pairs that satisfy Pálfy's inequality. Additionally, for a fixed positive integer $n$, the number of distinct graph orders for which exactly $n$ component size pairs satisfy Pálfy's inequality is shown.

Theorems & Definitions (5)

• Lemma 1
• proof
• Lemma 2
• proof
• proof