Editing Turán's theorem (section)

==Mantel's theorem==
The special case of Turán's theorem for <math>r=2</math> is Mantel's theorem: The maximum number of edges in an <math>n</math>-vertex [[triangle-free graph]] is <math>\lfloor n^2/4 \rfloor.</math>{{r|mantel}} In other words, one must delete nearly half of the edges in <math>K_n</math> to obtain a triangle-free graph.

A strengthened form of Mantel's theorem states that any Hamiltonian graph with at least <math>n^2/4</math> edges must either be the [[complete bipartite graph]] <math>K_{n/2,n/2}</math> or it must be [[pancyclic graph|pancyclic]]: not only does it contain a triangle, it must also contain cycles of all other possible lengths up to the number of vertices in the graph.{{r|bondy}}

Another strengthening of Mantel's theorem states that the edges of every <math>n</math>-vertex graph may be covered by at most <math>\lfloor n^2/4 \rfloor</math> [[clique (graph theory)|cliques]] which are either edges or triangles. As a corollary, the graph's [[Intersection number (graph theory)|intersection number]] (the minimum number of cliques needed to cover all its edges) is at most <math>\lfloor n^2/4 \rfloor</math>.{{r|egp}}