7.9 证明： χ ( G − e ) = min ⁡ { χ ( G ) , χ ( G ⋅ e ) } \chi(G-e) = \min \{\chi(G),\chi(G\cdot e) \} χ(G−e)=min{χ(G),χ(G⋅e)}

证：

1. 色多项式的递推公式：$P_k(G)=P_k(G-e)-P_k(G\cdot e)$​

2. χ ( G − e ) = min ⁡ { k ∣ P k ( G − e ) ≠ 0 } = min ⁡ { k ∣ P k ( G ) + P k ( G ⋅ e ) ≠ 0 } = min ⁡ { χ ( G ) , χ ( G ⋅ e ) } \begin{aligned} \chi(G-e) &= \min\{k | P_k(G-e) \ne 0 \} \\ &= \min\{k | P_k(G)+P_k(G\cdot e) \ne 0 \} \\ &= \min \{\chi(G),\chi(G\cdot e) \} \end{aligned} χ(G−e)​=min{k∣Pk​(G−e)=0}=min{k∣Pk​(G)+Pk​(G⋅e)=0}=min{χ(G),χ(G⋅e)}​