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I am currently a PhD student focusing in Functional Analysis, but I have to take a course in Graph Theory this semester.

So, the problem I have with theorems and proofs in Graph Theory is that they all just seem so arbitrary and random. They all look like a randomly found algorithm, which just happens to work. In functional analysis, you don't have proofs like this - here, you proofs are a logical consequence of applying former knowledge (example: the proof for the Hahn-Banach Theorem).

To name an example in graph theory, let's take the proof for the theorem that if $G$ is a graph of order $n$, and $|E(G)| \le n-2$, then we can find a packing with a cyclical permutation.

The proof - like lots of proofs in Graph Theory - relies on induction. We first show it for $n=3, 4$ and then go to the second step of the induction. Here, we analyze two cases: if $G$ has two isolated vertices, and if $G$ has no isolated vertices at all. And then what we do is we remove certain vertices and get a permutation $\sigma'$ from our inductive assumption, which can be extended by adding a transposition concerning the removed vertices. The problem I have here is that for each different case we extend $\sigma'$ to $\sigma$ in a slightly different way, which seems very arbitrary and random for me. But hey, "it works".

And I feel like that about all of the proofs and theorems in Graph Theory. I simply don't see why we choose this specific algorithm to prove a theorem, and what would happen if we would choose another one. This is especially difficult to show, because you would need to spend lots of time drawing graphs and all the different examples in order to see why it might not work.

Is there another way to learn proofs in Graph Theory, especially if you come from a purely functional analysis background? Right now, it really feels like rote learning for me.

anon
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    You're comparing a fundamental result of functional analysis (the Hahn-Banach theorem) with an example from graph theory that's so obscure that I've never even heard of it despite being a graph theorist. Of course the proof is going to be arbitrary and random: the Hahn-Banach theorem has the advantage of generations of mathematicians polishing both the proof itself and the foundations leading up to it to match. – Misha Lavrov May 27 '24 at 15:20
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    (Or, actually, if the claim you're making is equivalent to "every graph with at least $\binom n2 - n + 2$ edges has a Hamiltonian cycle" - it's not 100% clear to me what a "packing with a cyclical permutation" is, but it might mean that - then it does have a very different proof that's a logical consequence of applying former knowledge, with no casework required.) – Misha Lavrov May 27 '24 at 15:25
  • I would read all of the answers here if you haven't already. In my experience with not-hugely-advanced combinatorial maths, there are often one or two "clever ideas" in a proof which are really all you need to learn, then the rest of the proof writes itself. I think your experience also depends a lot on how clearly the ideas are presented to you/whether they're presented in a style that works for you. (It's funny you mention Hahn-Banach - I always struggled hugely to remember the obscure calculation you have to do to extend the functional!) – Izaak van Dongen May 28 '24 at 11:19
  • Not enough for an answer, but I think this is just what happens when you're new to a subject. If you kept going with graph theory, eventually you'd find you have intuition behind the important algorithms, and things wouldn't feel random anymore. – user1149748 May 29 '24 at 17:25
  • Even in the standard proof of the Hahn-Banach theorem, extending from the real case to the complex case involves an "algorithm" which at first might seem "random" and "just happens to work". – Daniel Schepler May 29 '24 at 19:12
  • @DanielSchepler I wouldn't say there is any algorithm involved at all, not at least in the version of the proof for complex numbers which I've learned (which was about applying the Hahn-Banach Theorem for real numbers for the direct value of a complex number, I don't remember the details but there was definitely not an algorithm involved and it was just very straight up calculations) – anon May 29 '24 at 20:37

2 Answers2

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I'm writing this as an answer because it didn't fit as a comment.

Historically combinatorics was thought of as a field lacking abstract structure, that is until Gian Carlo Rota's papers "On the foundations of combinatorial theory". Reading the introduction to the linked paper is worthwhile, as it is addressing a concern similar to yours. Graph theory is in a certain sense different from algebraic or analytical fields because the foundational structure theorems are very recent and very difficult to prove. For example, the Robertson-Seymour theorem took 20 years to prove and was concluded only in 2004.

Graph Theory is huge and full of (open) problems that look tame, but aren't. However, there are some common strategies. Typical is divide-and-conquer or/and reduction via contraction/deletion, the latter being what you are describing. That you have to then look at some special cases and apply different reasoning case by case is almost expected since you are dealing with discrete structures that often (to the best of our knowledge) do not have a "continuous thread" tying them together.

Learning Graph Theory is not different from other areas of mathematics - if you want to understand it well, you have to understand the motivation of the problems that you are trying to solve and the tools that are available to you. Most importantly, it takes time.

However, learning proofs specific to your course is a question for your lecturer. Any course can be more or less structured - if it is a hodgepodge of terminology, definitions, problems and tools that you have to absorb in 12 weeks, sometimes you don't have much choice.

My recommendation is that you talk to the lecturer, and if their response is not satisfactory, talk to your PhD advisor about the difficulties that you're having and how to resolve them.

Randy Marsh
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There is a huge misconception here that functional analysis is just about logical consequences of applying former knowledge. The exercises you are given in some university course in this area may be like this, but that does not imply anything about functional analysis in general. There is a reason that you are kind of more likely to see more ad-hoc techniques in discrete mathematics than analysis in undergraduate-level courses, and that is because the quantifier complexity of relevant concepts tends to be higher in analysis and so harder problems tend to be too hard for undergraduate students. In fact, for discrete mathematics (e.g. graph theory), being able to play around with concrete examples in order to get a grasp of a problem is what makes problems in this area more accessible to students.

In fact, your statement that graph theory feels like "rote learning" unlike functional analysis simply furnishes evidence that all the "functional analysis" that you have been doing so far is actually simple, and that your exposure to "graph theory" finally shows you what mathematics truly is like, where there are lots of possible paths to explore and often no clear indication as to which leads to anything useful.

user21820
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    So you say the Math I've been doing and I've been exposed to was pretty easy so far? I kind of get from where you are coming from. Initially, when I read your comment, I got pretty upset, but the more I think about it the more it unfortunately makes sense. Right now, the specific topic I am working on in functional analysis does not have this neat "linear" understanding, because it just got too advanced and nuanced and many cases need to be checked separately.

    So I guess that's it, I am just bad at Math if these kind of proofs are too hard and just got lucky to get into a PhD program.

     – anon May 28 '24 at 16:28
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    @anon Now you talk like a real PhD student. Just you wait until you're just lucky to get a postdoc. – Ainsley H. May 28 '24 at 17:25
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    @anon: Being bad at mathematics is not an intrinsic property. The vast majority of mathematicians are good at mathematics because of effort and time, and maybe a bit of good guidance from their teachers or advisors. You clearly seem to like mathematics, and that is one of the most important factors in giving you motivation to put in the necessary time and effort. When you reach 10000 hr of consistent effort, you would perhaps recall what I said and think that it was hard but not impossible after all. That said, if life needs you elsewhere, do not be afraid to go. Mathematics is not all. =) – user21820 May 29 '24 at 04:31