# Alternating and symmetric groups with Eulerian generating graph

## TL;DR

This paper studies the generating graph of symmetric and alternating groups, characterizing vertices with even degree and conditions under which the graph is Eulerian, revealing new structural properties of these groups.

## Contribution

It provides a complete characterization of vertices with even degree and establishes criteria for the generating graph to be Eulerian in symmetric and alternating groups.

## Key findings

- Vertices with even degree are explicitly identified.
- The generating graph is Eulerian if and only if certain prime conditions are not met.
- Conditions depend on the primality and congruence of n and n-1.

## Abstract

Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $\Gamma(G)$ when $G$ is an alternating group or a symmetric group. In particular, we determine the vertices of $\Gamma(G)$ having even degree and show that $\Gamma(G)$ is Eulerian if and only if $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4.