Let $G _ { 1 } = \left( S _ { 1 } , A _ { 1 } \right)$ and $G _ { 2 } = \left( S _ { 2 } , A _ { 2 } \right)$ be two non-empty graphs such that $S _ { 1 }$ and $S _ { 2 }$ are disjoint, that is, such that $S _ { 1 } \cap S _ { 2 } = \varnothing$. Let $s _ { 1 } \in S _ { 1 }$ and let $s _ { 2 } \in S _ { 2 }$.
We define the graph $G = ( S , A )$ with $S = S _ { 1 } \cup S _ { 2 }$ and $A = A _ { 1 } \cup A _ { 2 } \cup \left\{ \left\{ s _ { 1 } , s _ { 2 } \right\} \right\}$.
Show that :
$$\chi _ { G } = \chi _ { G _ { 1 } } \times \chi _ { G _ { 2 } } - \chi _ { G _ { 1 } \backslash s _ { 1 } } \times \chi _ { G _ { 2 } \backslash s _ { 2 } }$$