The edge-flipping group of a graph (original) (raw)

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 ϵ∈ E and change the colors of all adjacent edges of ϵ. 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 W_E(X) of X. This paper shows that if X has at least three vertices, W_E(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.

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