Reposting my comment from /r/math: OK, section 4.2 makes no sense to me. So they have this set Q, right? Which is the set of plane graphs that can be written G

Reposting my comment from /r/math: OK, section 4.2 makes no sense to me. So they have this set Q, right? Which is the set of plane graphs that can be written G=A*B*C*D, where * denotes joining at a cut vertex. Their goal is to show that every G∈Q can be 4-colored, and then use that to argue that a nonzero fraction of plane graphs can be 4-colored, and then argue (via the generating functions and asymptotics) that this means that they all can be. Except... if they can show that every G∈Q can be 4-colored, then all of this argument with generating functions and asymptotics is unnecessary, right? Because surely G can be 4-colored iff A, B, C, and D all can; and since A, B, C, and D can be any plane graph, that would be the end of it -- if you have a plane graph A and you want to 4-color it, all you'd have to do is stick a 3-path hanging off the end of it to put it in Q (indeed, in the subset they call \overline{Q}), and then 4-color that. The rest of the argument would appear to be unnecessary! Meanwhile, their argument for why graphs in Q can be 4-colored doesn't seem to make any sense to me. It's inductive, but they haven't written it in a clear manner that makes it exactly apparent how they inductive hypothesis is used. I honestly do not see how they are justifying the statement "It follows from the Induction Hypothesis that S_1 is 4-colorable" -- it doesn't look like any work is being done, it all looks circular to me! Am I missing something here?