On the period-two-property of the majority operator in infinite graphs

A self-mapping M : X → X M:X → X of a nonempty set X X has the Period-Two-Property (p2p) if M 2 x = x {M^2}x = x holds for every M M -periodic point x ∈ X x ∈ X . Let X X be the set of all { 0 , 1 } \{ 0,1\} -labelings x : V → { 0 , 1 } x:V → \{ 0,1\} of the set of vertices V V of a locally finite connected graph G G . For x ∈ X x ∈ X let M x ∈ X Mx ∈ X label v ∈ V v ∈ V by the majority bit that x x applies to its neighbors, retaining