https://dspace.library.uu.nl/bitstream/handle/1874/842/full.pdf?sequence=1
At the bottom of page 115 of the above it states that'By planarity one can verify that every vertex v of G has at least one neighbor u such that u and v have exactly two neighbors in common.'
I have been able to prove this if the degree of the vertex is 5, but how would I go about proving it for all vertices in the graph?