Zykov symmetrization

Zykov symmetrization replaces one of two nonadjacent vertices by a clone of the other. Cloning the vertex of larger degree does not decrease the edge count and does not create a larger clique. Iteration reduces the extremal problem for clique-free graphs to complete multipartite graphs.

For each real $c$, some maximizer of $e(G)-c{k}_3(G)$ among graphs on $n$ vertices is complete multipartite graph. Among maximizers maximize ${\sum }_{v}d(v{)}^2$. Cloning either of two nonadjacent vertices preserves the objective, since maximality forces their local contributions $d(v)-ce(G[N(v)])$ to agree. If their vertex neighbourhoods differ, the sum of the two changes in the squared-degree of a vertex objective is twice the size of their symmetric difference, a contradiction. Nonadjacency therefore partitions the vertices into classes with identical vertex neighbourhoods.