A graph is Zm-well-covered if |I| ≡ |J| (mod m), for all I, J maximal independent sets in V(G). A graph G is strongly Zm-well-covered if G is a Zm-well-covered graph and G\{e} is Zm-well-covered, ∀ e ∈ E(G). A graph G is Zm-well-covered if G is Zm-well-covered and G\{v} is Zm-well-covered, ∀ e ∈ V(G). We prove that K1 and K² are the only 1-Zm-well-covered graphs with girth > 6. They are also the only ones with girth > and strongly Zm-well-covered. We show a necessary and sufficient condition for the lexicographic product of graphs to be a Zm-well-covered one and some properties for the cartesian product of cycles.
Graph theory; independent sets in graphs; graph products
