A proper colouring using exactly $k$ colours first partitions the vertices into $k$ nonempty independent colour classes and then injectively assigns $k$ of the $x$ named colours to those classes. These choices number
$$
\left\{\begin{matrix}n\\k\end{matrix}\right\}_Gx^{\underline k}.
$$
Summing over $k$ counts every proper colouring exactly once; the expression is the <chromatic polynomial> $\chi_G(x)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 145 / 2 / f / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-145/2/f/solution