# Minimal and maximal degrees in critical graphs

Let $$G=(V,E)$$ be a finite simple undirected graph. We say that $$G$$ is critical if for all $$v\in V$$ we have $$\chi(G\setminus\{v\}) < \chi(G).$$ By $$\Delta(G)$$ and $$\delta(G)$$ we denote the maximum and minimum degrees of $$G$$, respectively.

Let $${\cal C}$$ be the set of all finite critical graphs $$G=(V,E)$$ with $$V \subseteq \mathbb{N}$$ and $$|V|>1$$. Is the set $$\big\{\frac{\Delta(G)}{\delta(G)}:G \in{\cal C}\big\}\subseteq \mathbb{Q}_{\geq 0}$$ bounded?

(Note that $$G$$ critical and $$|V(G)|>1$$ implies $$\delta(G) > 0$$.)

No, it is unbounded: for the wheel graph $$W_n$$ of even order $$n$$, we have $$\Delta(G)=n-1$$, $$\delta(G)=3$$, $$\chi(G)=4$$, and $$\chi(G\setminus\{v\})=3$$ for any vertex $$v$$.