The [complete graph][1] with $k+1$  vertices (which is strongly $k$-regular) has degree $k$ and chromatic number $k+1$. On the other hand, by [Brook's theorem][2] this is the maximum possible value for the chromatic number of a $k$-regular graph, so $$\limsup_k \frac{c_k}{k} = \lim_k \frac{c_k}{k}=1.$$   


  [1]: https://en.wikipedia.org/wiki/Complete_graph
  [2]: https://en.wikipedia.org/wiki/Brooks%27_theorem