Challenging Graph theory Problems?

Question

Continuing from this , I just want to know some Graph theory books which also have good (read:tough) problems.

I am asking this because I have noticed that with a lot of 'college' books that the excercises are usually not at all challenging,

For instance, Thomas' Calculus; while the books contents itself are really cool and well explained, I could never really know if I really understood the material well, because the problems were usually not very tough, pretty standard is what I mean (although I have only read the first $5-6$ chapters) and so if I tried some (much)harder problems, only after doing standard excercises (like say Putnam), I'd probably fail.

I suppose I could just try past Olympiad problems after reading some chapter, but it would really be very tough to find problems that involve some specific stuff (say I read about Planar graphs, then It would take me a veryyyyyy long time just to try and find a couple Olympiad problems on Planar graphs, and I might not even be able to recognize them just yet).

I hope this is not a duplicate, as I have tried my best to make myself clear as to what I am actually searching for.

Thank you!

EDIT: I have glanced through Diestel though, and some questions seemed tough but I am just a beginner, so I am not sure. So, if anyone has done Diestel, does that work?

This isn't specific to graph theory, but there might be some useful content in MIT's 18.211 Combinatorial Analysis. — Carl Schildkraut, Aug 11 '21 at 21:26
https://www.amazon.com/Combinatorial-Problems-Exercises-Chelsea-Publishing/dp/0821842625 has a large number of graph theory problems . — Asinomás, Aug 11 '21 at 21:27
@CarlSchildkraut I don’t suppose there are any lectures or reading materials from that course available? The problems seem like they are the kind of thing I would want to learn about — FShrike, Aug 11 '21 at 21:29
@FShrike The lectures don't seem to be up (nor can I find lecture notes) but there's a link to the textbook (A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory by Miklos Bona), if that would be helpful. — Carl Schildkraut, Aug 11 '21 at 23:08

score 2 · Accepted Answer · answered Aug 12 '21 at 02:29

Here are a few options:

Diestel's Graph Theory, which you already have. In my opinion, it is solid preparation for (let's say) olympiad level graph theory, but easier than many olympiad problems. Many of the harder problems prove things that I'd expect a graduate student in combinatorics to already be familiar with and have in their toolbox - rather than be challenged by.

West's Introduction to Graph Theory has many many more exercises than Diestel. If you are specifically on the lookout for challenging problems, you will find some here. Many of the harder problems have the flavor of lemmas you would need to prove in actual graph theory research (because that's where they come from).

Both Diestel and West have many routine exercises (which is also useful) and many harder ones, which are marked with special symbols next to the exercise numbers.

Alon and Spencer's Probabilistic Method has exercises notorious for being difficult, many of them related to graph theory (but with a specific focus that can be seen from the title of the book).

You might also consider looking at problem-solving textbooks specifically, some of which might have a graph theory chapter, but I have no concrete recommendations.

score 0 · Answer 2 · answered Aug 11 '21 at 21:20

0

Here's one: Try to write a program that solves a Rubik's Cube in the fewest number of rotations possible. I believe the maximum number of moves is $20.$ So like, each node/vertex is a particular configuration/state of the cube, and edges join one state to another. I made some progress on this problem using some DFS algorithms, but didn't quite complete it. I don't think computing power gets you there via brute force DFS, so you have to come up with some shortcuts... Kind of similar to chess programming.

answered Aug 11 '21 at 21:20

Adam Rubinson

24,300

That's a fun problem, but it's not a suggestion for a book on graph theory. It has some practical challenges, but I'm not sure I'd call it tough in the sense of olympiad problems. (By the way, you can run two breadth-first searches simultaneously, one from the mixed state and another from the solved state, and see when they meet.) – Kyle Miller Aug 12 '21 at 03:25
Yes but even that takes a long time. You need something shorter – Adam Rubinson Aug 12 '21 at 09:05

Challenging Graph theory Problems?

2 Answers2