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Let $G= (V,E)$ be a graph such that:
$E = E_1 \cup E_2$
$G_1 = (V, E_1)$ is planar
$G_2 = (V, E_2)$ is forest
Proof that chromatic number is $< 9$

My observations

Each planar graph has chromatic number $\le 4$. Also each forest has chromatic number $2$ if $|V| > 1$

But how can I connect that and have convince about chromatic number $<9$?

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    $\begingroup$ Note that $4\times 2=8<9$. Can you see how to finish? $\endgroup$ – user10354138 May 16 at 16:35
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Given colorings $C_1$ of $G_1$ and $C_2$ of $G_2$, construct a new coloring $C_1 \times C_2$ where $C_1 \times C_2(v) = ( C_1(v),C_2(v))$. This is a valid coloring of the graph (check this!). Since there are at most 4 colors in $C_1$ and at most 2 in $C_2$, there are at most 8 in $C_1 \times C_2$.

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  • $\begingroup$ Do you mean coloring in use of colors defined by $2$ numbers? $\endgroup$ – trolley May 16 at 17:00
  • $\begingroup$ I don't understand your question. The new coloring is given by ordered pairs of the first coloring and the second coloring. $\endgroup$ – user113102 May 16 at 17:55
  • $\begingroup$ I was asking exactly about that @user113102 $\endgroup$ – trolley May 16 at 17:58

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