# graph theory - maximal graph without cycles and maximal planar graph without triangles

Let there be $$G = (V,E_1\cup E_2)$$ such that $$(V,E_1)$$ is a planar graph without triangles, and $$(V,E_2)$$ is without circles, show that $$G$$ is $$6$$-colorable

Can I prove here that $$E_1$$ and $$E_2$$ has at least one edge that is inside both of them when the vertices are $$v_1,v_2,v_3,\dots,v_n$$, I couldn't find why this is wrong, can someone please show it using the pigeon hole principle.

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– Pedro
Aug 5, 2022 at 9:12