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 $\delta(G) > 0$.)


1 Answer 1


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 $v$.


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