Reasons for the importance of planarity and colorability? Could it have been foreseen that - exemplarily - planarity and colorability would turn out to be such important concepts in graph theory (there's almost no textbook on graphs without two chapters devoted to these concepts), or did time have to come and show it? The latter might imply that it's just a historical and evolutionary incident because there are loads of conceivable x-arities, y-abilities and other graph properties. (Maybe the importance of planarity and colorability just has to do with the contingent fact that we live on a (locally) two-dimensional plane and our need of maps?)

But maybe there are more objective reasons internal to mathematics that are formulable?

Related questions: Why-are-planar-graphs-so-exceptional; generalizations-of-planar-graphs;why-is-edge-coloring-less-interesting-than-vertex-coloring
 A: The main feature of Kurtowski's theorem on planar graphs at the time was that it connected two seemingly unrelated aspects of graphs, one topological in nature and the other combinatorial. Since then different authors have proved many theorems about planar graphs and they are now a fairly understood family. So I guess, the reasons for the importance of planarity have changed a bit with time.
Since many hard problems simplify a lot in the planar case this makes them a very good pedagogical tool to introduce when teaching/motivating various results in graph theory. As far as I know, planar graphs where the main reason to support Tutte's flow conjectures, for example. Other problems/conjectures where the planar case makes an interesting toy model are Fleischner's conjecture, circuit decomposition (Hajos), strong perfect graph conjecture, strong embedding conjecture, strong cycle double cover conjecture etc.
Part of motivation comes from topological graph theory, too. If you are interested in graphs as discretized version of surfaces, the planar case is probably the first you want to look into. Here the subject may jump to fields other than graph theory, though, and you may start to ask about properties under scalings and subdivisions (think of dimer models for example), but then there are more reasons that enter the picture for why the planar case is interesting, such as conformality for instance.
A: I agree with the points of David Eppstein's answer on the important of planarity. I can add also my answer to a similar problem. We still need a good answer on why colorability is so important. As Tim Gowers said it is studied in many areas of graph theory and in also outside graph theory. Colorability is computationally intractible yet it is mathematically more tractable compared to other computational intractible questions like Hamiltonianity.
Let me try to suggest some answers on why graph colorability is important (I think it is a tentative and partial list):
1) It is a very easy to define and a very natural concept.
2) It is related to real life questions like scheduling and map coloring.
3) It is related to an important algebraic notion (that came from its study) the chromatic polynomial.
4) It led to many important generalizations. Like chosability.
A: A few reasons for the importance of planarity having little to do with the need for maps:


*

*A matroid is both graphic and co-graphic if and only if it is the graphic matroid of a planar graph

*Planar graphs are the graphs with Colin de Verdiere invariant ≤ 3. As such they form a sequence with the trees, outerplanar graphs, planar graphs, and linklessly embeddable graphs.

*The graphs of three-dimensional convex polyhedra are exactly the 3-connected planar graphs (Steinitz's theorem).

*A minor-closed graph family has bounded treewidth if and only if it does not include all the planar graphs.
A: I think that planarity figures more heavily in typical undergraduate courses than its importance at research level would warrant. I'm not saying that it's not important at research level, but I do think that it is noticeably less important than graph colouring. In particular, there are whole areas of graph theory where planarity doesn't come up, whereas colouring is fairly ubiquitous.
Why the emphasis at the undergraduate level? This is easily explained: the 4-colour theorem is a very famous result, and the 5-colour theorem has a nice proof that is suitable for an undergraduate course. There is nothing wrong with this at all: not everything that goes into an undergraduate course is there as direct training for research.
A: Mesh Analysis a method used to solve circuits only applies to planar circuits.
A: There is the circle packing theorem, every connected simple planar graph is isomorphic to a circle packing.
