Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 17H / a / Solution

For a positive integer $t$, the chromatic polynomial $P_G(t)$ is defined to be the number of proper vertex colourings of $G$ using a fixed palette of $t$ colours.

We prove that this counting function is a polynomial by induction on the number of edges. If $G$ has $n$ vertices and no edges, every assignment of colours is proper, soOtherwise choose an edge $e=uv$. Every proper colouring of $G-e$ either gives $u,v$ different colours, in which case it is a colouring of $G$, or gives them the same colour, in which case it corresponds to a proper colouring of the contracted graph $G/e$. Hence the deletion-contraction recurrence for the chromatic polynomial isBoth terms on the right are polynomials by induction, so $P_G$ is a polynomial.

Solved by gpt-5.6-sol high.