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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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What do you mean by "order"? – draks ... Mar 9 '12 at 11:03
I mean number of vertices. I'll edit the question. – mrb Mar 9 '12 at 11:10
With that many edges, it might be easier to characterize the complement. – Gerry Myerson Mar 9 '12 at 11:33
By $\chi(G)$ you mean the chromatic number or the Euler charateristic? – draks ... Mar 9 '12 at 11:38
I think order is standard and $\chi(G)$ is standard for chromatic number, and Euler characteristic couldn't possibly be n - 2, could it? – Graphth Mar 9 '12 at 14:59

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