For any graph $G=(V,E)$, the coloring number $\text{Col}(G)$ is defined to be the smallest cardinal $\kappa$ such that there is a well-ordering $\leq$ on $V$ such that for every vertex $v\in V$ we have $$|N(v) \cap \{w\in V: w \leq v\}| \leq \kappa,$$ where $N(v)=\{x\in V:\{x,v\}\in E(G)\}$.


It is known that $\chi(G) \leq \text{Col}(G)$ for all graphs $G$.


The Hadwiger number $\eta(G)$ of a finite graph $G$ is the largest integer $n\in\mathbb{N}$ such that $K\_n$ is a minor of $G$. [Hadwiger's conjecture](http://en.wikipedia.org/wiki/Hadwiger_conjecture_%28graph_theory%29) states that $\chi(G)\leq \eta(G)$ for all finite graphs.


**Question**: Is there a finite graph $G$ such that $\eta(G) < \text{Col}(G)$?



Hadwiger's conjecture for coloring number instead of chromatic number
Take the complete 3-partite graph $K\_{n,n,n}$. Coloring number $2n+1$ (with the correct +1 definition). Any $K\_{2n+1}$ minor would have to have at least $n+1$ singleton branch sets, i.e. there would be a $K\_{n+1}$ subgraph (which is not there for $n\ge 3$).


