Abstract.
Given a graph G with n vertices and stability number α(G), Turán's Theorem gives a lower bound on the number of edges in G. Furthermore, Turán 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 Turán's Theorem to the q-stability number, for q>2.
Access this article
Rent this article via DeepDyve
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: July 21, 1999 Final version received: August 22, 2000
Present address: GERAD, 3000 ch. de la Cote-Ste-Catherine, Montreal, Quebec H3T 2A7, Canada. e-mail: Alain.Hertz@gerad.ca
Rights and permissions
About this article
Cite this article
Gerber, M., Hansen, P. & Hertz, A. Extension of Turán's Theorem to the 2-Stability Number. Graphs Comb 18, 479–489 (2002). https://doi.org/10.1007/s003730200034
Issue Date:
DOI: https://doi.org/10.1007/s003730200034