Proper vertex colourings and chromatic number

Let $G=(V,E)$ be a finite simple graph (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets) and let $k\in\mathbb N$. A proper $k$-vertex-colouring is a function

$c:V\longrightarrow k$

such that $c(u)\ne c(v)$ whenever $\{u,v\}\in E$. Its fibres are the colour classes. The graph is $k$-colourable when such a function exists.

The chromatic number of $G$ is

$\chi(G):=\min\{\,k\in\mathbb N:G\text{ is }k\text{-colourable}\,\}.$

This minimum exists. Since $V$ is finite, there is a bijection $b:V\to |V|$ (The cardinality $\lvert A\rvert$ of a finite set), and $b$ is a proper $|V|$-colouring because adjacent vertices are distinct. The displayed set of admissible natural numbers is therefore nonempty, so it has a least element by The well-ordering principle.

The null graph has $V=\varnothing$. Its unique empty function $\varnothing\to0$ is a proper $0$-colouring, so $\chi(G)=0$. Conversely, a nonnull graph has no function from its nonempty vertex set to $0=\varnothing$, and hence has positive chromatic number. Colours are labels only: composing a proper colouring with a bijection of its colour set changes no adjacency condition.