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Attaching a new leaf to a graph multiplies its chromatic polynomial by $t-1$, because after colouring the old graph the leaf may receive any colour except its neighbour's.

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.

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 numberSumming over $k$ counts every proper colouring exactly once; the expression is the chromatic polynomial ${\chi }_G(x)$.

A finite graph is bipartite exactly when it has a proper colouring with two colours. Equivalently,for its chromatic polynomial.