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- #1

- Mar 10, 2012

- 834

"A strengthened form of Mantel's theorem states that any graph with at least $n^2/4$ edges must either be a complete bipartite graph $K_{n/2,n/2}$ or it must be pancyclic, i.e, not only does it contain a triangle, it must also contain cycles of all other possible lengths up to the number of vertices."

Now a special case of this is:

A non-bipartite graph $G$ with $|E(G)|\geq n^2/4$, where $n=|V(G)|$, has a Hamiltonian Cycle.

But if we consider the $10$ vertex graph $G$ which has a $9$-clique and an isolated vertex then we see that $|E(G)|=36>10^2/4$ but $G$ doesn't have a Hamiltonian cycle.

What has gone wrong here? What am I missing?