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This is quite a soft question. I'm looking for any properties that a graph $G$ on $n$ vertices satisfying the following conditions might have:

  1. $\chi(G)=n-2$
  2. $|E(G)|>(n^2-3n+6)/2$

Clearly, for example, $n$ must be greater than 3, the clique number of $G$ must be less than or equal to $n-2$, and the density of the graph must be greater than: $\displaystyle\frac{n^2-3n+6}{n^2-n}$.

Are there any nice characterizations of such graphs?

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    I think order is standard and $\chi(G)$ is standard for chromatic number, and Euler characteristic couldn't possibly be n - 2, could it?2012-03-09

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