Chromatic polynomial of a cycle ($p_{C_{\ell }}(x)=(x-1{)}^{\ell }+(-1{)}^{\ell }(x-1)$)

For a cycle graph of length $\ell \ge 3$, the chromatic polynomial is $p_{C_{\ell }}(x)=(x-1{)}^{\ell }+(-1{)}^{\ell }(x-1)$. The graph triangle gives the base case $x(x-1)(x-2)$. For $\ell \ge 4$, the deletion-contraction recurrence for the chromatic polynomial gives $p_{C_{\ell }}(x)=x(x-1{)}^{\ell -1}-p_{C_{\ell -1}}(x)$, since deleting an edge leaves a path graph and contracting it gives the shorter cycle graph. Substitution proves the formula by mathematical induction.