The flipping group of a line graph
Hau-Wen Huang, Chih-wen Weng · 2008
Let X be a simple connected graph with n vertices and m edges. Every vertex of X is assigned either black state or white state. We move by selecting a vertex v of X having black state and then change the states of all neighbors of v. This is the flipping puzzle on X and it corresponds to a group action. We referred this group to the flipping group of X. In this paper, we are mainly concerned about the flipping group on the line graph L(X) of X. We show that the flipping group of L(X) is isomorphic to a semidirect product of (Z/2Z) k and the symmetric group Sn, 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.