Thanks for the many great suggestions. Reading all these has caused me to think that I could potentially structure the course in such a way as to introduce the widest number of adjectives that can precede "graph theory" or "combinatorics". For example, I see in the topics presented here: enumerative, extremal, geometric, computational, probabilistic, algebraic, and constructive (for lack of a better word - I'm referring to things like designs). As a sort of subquestion restricted to the comments, what other adjectives might I attempt to incorporate?

I started to call this "absorption", but that is not quite correct (absorption is a relationship between two operations). Still, perhaps that name will jog someone's memory.

Sorry for the unintended bump. I meant only to test the rollback feature.

A fact (not proven) from Diestel comes to mind: The random graph with probability of edge inclusion equal to $(1 + \epsilon)(\log n)n^{-1}$ is almost surely Hamiltonian.

I think your only alternative is to present the "magic" differentiation rules with no justification. It is already common for students to have a black-box view of mathematics; I don't think you want to encourage it. Perhaps you want to begin with the definition via limits and then derive the rules from there. Emphasize to your students that "Why didn't we just use the rule from the start?" is not a valid question. The rule is a consequence of the definition, not a self-evident truth.

I am a fool when it comes to geometry, but I think a solution to your general problem of finding 3 points forming a triangle with specified angles would imply that an arbitrary angle could be trisected, which is impossible.

en.wikipedia.org/wiki/Compass_and_straightedge_constructions In particular, you'll find some key constructions that are impossible.

Just a minor point: Your function appears to be defined on $V(\Gamma) \times V(\Gamma)$, as $\phi_T$ only has a meaningful output when given two vertices.