Given a positive integer $n\in \mathbb{N}$, is there a positive integer $k\in{\mathbb N}$ such that



> 
> for every finite, simple, undirected graph $G$ with $\Delta(G) = n$ we have $\chi(G) \leq k$
> 
> 
> 


?



Maximal degree and chromatic number
$k=n+1$. For the coloring with $n+1$ colors, use induction on the number of vertices. 


