Not only do interesting questions arise by considering the special class of planar graphs but additional special issues arise when one considers a specific plane drawing of a planar graph. This is because when a graph is drawn in the plane it becomes possible to order or number, say for example, the edges at a particular vertex of the graph, in an organized consistent way.
One example of results which started from this reality for plane graphs is this paper of Grunbaum and Motzkin: B. Grünbaum and T. S. Motzkin , The number of hexagons and the simplicity of geodesics on certain polyhedra. Canad. J. Math. 15 (1963), pp. 744–751. In this paper the following is explored (along with results about what today are called fullerenes). Suppose one has a 3-valent plane graph. If one picks any edge and moves along that edge (in either direction), when one gets to a vertex one has the choice of going left or right and moving on to another edge. Suppose one goes left, and at the next vertex in one’s “traversal” one goes right, alternating left, and right. Grunbaum and Motzkin refer to this as left-right path, and they prove that for a plane 3-valent graph with each of its faces having a multiple of 3 for its number of sides, that these left-right paths (starting on any edge) always generate “simple circuits.”
I was able to extend this result in several directions. One such idea applies to the graph that serendipitously appears as the diagram that starts this “thread.” This graph is plane 4-valent, so when one moves along an edge and one gets to a vertex one might always choose to take the middle edge. For the graph above, when one does this, the graph breaks up into the union of simple closed curves. (It is easy enough to generate such graphs. Plop down a bunch of simple closed curves which cut each other when they meet transversely and think of the result as a 4-valent graph.) What are sufficient conditions for this (moving along middle edges to generate simple closed curves) to happen? Perhaps surprisingly, one such condition is that each of the faces of the plane 4-valent graph have a number of sides which is is a multiple of 3.
What sometimes happens, which is an interesting phenomenon in its own right, is that one generates an eulerian circuit by always choosing the middle edge in a 4-valent plane graph. So the theorem I mentioned just a moment ago can be interpreted to say that there is no knot projection which results in a 4-valent graph all of whose faces have a multiple of 3 as their number of sides. (Knots would be eulerian when one moves along a middle edge.) I like to think of left-right (more precisely, far left, far right) paths (which makes sense for 4-valent graphs, too) or take the middle edge in a 4-valent graph as arising because one has numbered the edges from left to right as one gets to a vertex with the numbers 1, 2, 3. (Similarly, for 3-valent or 5-valent plane graphs.) Now one can ask a large number of questions about the behavior paths which obey a certain “code.” Left-right paths (4-valent case) are the code 1,3 while take the middle edge is the 2 code. Once more, because of the graph being planar (plane) one gets interesting mathematical ideas, and I suspect there are many interesting results and new ideas to be obtained from this line of thinking.
So, yes, planar and plane graphs are exceptional.