Geodesic
On the Non-(p−1)-Partite Kp-Free Graphs
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 9-23

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We say that a graph G is maximal K_p-free if G does not contain K_p but if we add any new edge e ∈ E(G̅) to G, then the graph G + e contains K_p. We study the minimum and maximum size of non-(p − 1)-partite maximal K_p-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal K_p-free graphs of size m?
Keywords: extremal problems, maximal Kp-free graphs, Kp-saturated graphs 
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Amin, Kinnari; Faudree, Jill; Gould, Ronald J.; Sidorowicz, Elżbieta. On the Non-(p−1)-Partite Kp-Free Graphs. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 9-23. http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a1/