I have had very little contact with graph theory in the time since I graduated. As an undergraduate I used Robin Wilson's Introduction to Graph Theory, and I am now using it as the basis of my course. I remember enjoying the book in my youth, and finding it approachable, but I don't remember finding the material as straightforward as it now seems. (My students aren't finding it entirely straightforward, either, but that may be my fault.)
Why is this? I don't think I'm a better mathematician than I was 35 years ago. In terms of solving exam questions, I would not perform as I did when I was twenty. Even with practice, I am sure I could not get back to that level, and not only because I no longer value that kind of cleverness enough to put the effort in. I now have a much better general understanding of mathematics and how it all fits together, but I no longer have the ability to master detail that I once did.
Perhaps Wilson's book (which has gone through four more editions since my undergraduate days) has improved, but, with all due respect to its distinguished author, I doubt if it has really changed sufficiently to make a difference. (Pure mathematics doesn't change much: theorems that are proved generally remain proved, the Four-Colour Theorem notwithstanding.)
Learning mathematics takes time, and it has always astonished me how much better I understand material when I go back to it, months or years later, than when I first studied it. As John von Neumann is said to have told a student who complained that they didn't understand a piece of mathematics, "You don't understand mathematics, laddie, you get used to it." Even if I haven't looked at Philip Hall's Marriage Theorem, for example, for 35 years, the proof seems much simpler to me now than it did when I was first immersed in the subject area.
Perhaps I am misremembering my difficulties as a student: perhaps I didn't find it as difficult as I now remember it. Certainly I had little understanding of how an area of mathematics fitted together: my learning at University consisted of reading strings of definitions and theorems, with little idea where it was all going, making sure I understood each result before going on to the next one, until, perhaps, in the last lecture of the course the lecturer would say something like "and so we have now classified all Lie algebras" and I would suddenly find out what the point of it all had been. I now feel that I would have been a much more effective mathematician if I had read more superficially, skipping proofs until I understood the context, but since got good marks as an undergraduate I had no incentive to adopt what I now feel would have been a much better strategy.
But I think it is the case with mathematics, much more than with many other disciplines, that time is essential to understanding. Things we struggle with become much simpler when we return to them months later. This is why modularisation of mathematics studies is so pernicious. Examining students in the same semester as they have learned an advanced mathematics topic is, I feel, grossly unfair. It forces our exams to be superficial and makes it impossible to test deep understanding. At least, although my graph theory course finishes in December, the exam is not till May. I suspect my students don't like that, but they are likely to do much better than if they faced the same exam immediately after the final lecture.