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Francesco Polizzi
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The complete graph with $k+1$ vertices (which is strongly regular) has degree $k$ and chromatic number $k+1$, and by Brook's theorem this is the massimum possible value for a $k$-regular graph, so the $$\limsup_n \frac{c_k}{k} = 1.$$

Francesco Polizzi
  • 66.3k
  • 5
  • 180
  • 283