Extension of Turán's Theorem to the 2-Stability Number

Given a graph G with n vertices and stability number α(G), Turan's Theorem gives a lower bound on the number of edges in G. Furthermore, Turan has proved that the lower bound is only attained if G is the union of α(G) disjoint balanced cliques. We prove a similar result for the 2-stability number α2(G) of G, which is defined as the largest number of vertices in a 2-colorable subgraph of G. Given a graph G with n vertices and 2-stability number α2(G), we give a lower bound on the number of edges in G and characterize the graphs for which this bound is attained. These graphs are the union of isolated vertices and disjoint balanced cliques. We then derive lower bounds on the 2-stability number, and finally discuss the extension of Turan's Theorem to the q-stability number, for q>2.