On 2-periodic graphs of a certain graph operator

We deal with the graph operator $\overline{Pow₂}$ defined to be the complement of the square of a graph: $\overline{Pow₂}(G) = \overline{Pow₂(G)}$. Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph $K_{m,n}$ can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective geometries belong to and find infinitely many graphs also belonging to among generalized hypercubes.