After learning some very useful Graph Theory in this question (What is the correct term for cycles more fundamental than Simple Cycles?), I realized that in my work I have been considering a conjecture, which I would like to know if it is a theorem.
Has this been proven? For planar graphs, any edge of an undirected cyclic graph may participate in a maximum of two faces.
Can anyone confirm or correct me?
Yes, given a planar embedding of a planar graph, every edge is incident to at most two faces. Cyclicity of the graph doesn't really have anything to do with it.
The Jordan curve theorem is often quoted as justification for this statement, but exact details are often left out in most texts that I have seen, probably because the proof is rather quite fiddly and more topology than graph theory proper.
If you really want to see a proof written out, Combinatorial Optimization by Korte and Vygen has one. (Although even there it's rather sketchy, and also only proves the statement for 2-connected planar graphs, in which case every edge is incident to exactly two faces.)