The edge-flipping group of a graph

Abstract

Let X = (V, E) be a finite simple connected graph with it vertices and In edges A configuration is all assignment of one of the two colors, black or white, to each edge of X. A move applied to a configuration is to select a black edge c c E and change the colors of all adjacent edges off Given all Initial Configuration and a filial Configuration, try to find a sequence of moves that transforms the Initial Configuration into the final configuration This is rile edge-flipping puzzle oil X, and it corresponds to a group action This group is called the edge-flipping group W(E)(X) of X This paper shows that if X has at least three vertices. W(r)(X) is isomorphic to a semidirect product of (Z/2Z)(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 Z is the additive group of integers (C) 2009 Elsevier Ltd All rights reserved.