Let $c\_k$ be the maximum chromatic number that a $k$-regular graph can have. What is $\lim\sup\_{k\to\infty}\frac{c\_k}{k}$?



Maximum chromatic number of a $k$-regular graph
The [complete graph](https://en.wikipedia.org/wiki/Complete_graph) 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](https://en.wikipedia.org/wiki/Brooks%27_theorem) this is the maximum possible value for the chromatic number of a $k$-regular graph, so $$\limsup\_k \frac{c\_k}{k} = \lim\_k \frac{c\_k}{k}=1.$$ 


