# Mutually disjoint triangles in planar cubic graph

Let $$G$$ be a connected, planar graph for which every vertex has degree $$3$$ . Moreover, suppose there is a face of length at least $$12$$ .

Is it possible to construct an example of such $$G$$ whose only odd faces are six triangles, and for which no two such triangles share a common vertex?

(For my purposes I may assume there are no faces of length $$1$$ or $$2$$, in which case there must be an odd face of length at least 3. By the handshake lemma, there are an even number of, hence at least 2, such odd faces.)

• Triangular prism? – Zachary Hunter Dec 30 '18 at 6:02
• Thanks again. I've added that only 6 triangles are allowed. – Finallysignedup Dec 30 '18 at 6:08
• Why have you repeated your question? The solution given in the first can be easily adapted to this. – Daniel Mathias Dec 30 '18 at 6:10
• I've added some further restrictions, thanks. – Finallysignedup Dec 30 '18 at 6:20