The edge-flipping group of a graph

TL;DR

This paper characterizes the algebraic structure of the edge-flipping group associated with a finite simple connected graph, revealing it as a semidirect product of elementary abelian 2-groups and symmetric groups, depending on the graph's parameters.

Contribution

It provides a complete description of the edge-flipping group as a semidirect product, generalizing previous understandings of similar combinatorial group actions.

Findings

The edge-flipping group is isomorphic to a semidirect product of $( ext{Z}/2 ext{Z})^k$ and $S_n$.

The value of $k$ depends on the parity of the number of vertices $n$ and the graph's edges.

This structure holds for all connected graphs with at least three vertices.

Abstract

Let $X=(V,E)$ be a finite simple connected graph with $n$ vertices and $m$ edges. A configuration is an assignment of one of two colors, black or white, to each edge of $X.$ A move applied to a configuration is to select a black edge $\epsilon\in E$ and change the colors of all adjacent edges of $\epsilon.$ Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on $X,$ and it corresponds to a group action. This group is called the edge-flipping group $\mathbf{W}_E(X)$ of $X.$ This paper shows that if $X$ has at least three vertices, $\mathbf{W}_E(X)$ is isomorphic to a semidirect product of $(\mathbb{Z}/2\mathbb{Z})^k$ and the symmetric group $S_n$ of degree $n,$ where $k=(n-1)(m-n+1)$ if $n$ is odd, $k=(n-2)(m-n+1)$ if $n$ is even, and $\mathbb{Z}$ is the additive group of integers.