Apologies if the content of this post is not mathematical enough and is consequently in the wrong place. For context, I graduated with my masters in mathematics about 8/9 years ago and then went on to do other things, leaving maths behind. Recently I’ve got into learning Graph Theory (initially to help someone who was writing a module) and am having thoughts about taking up the torch again and doing a phd.
Now, I am able to learn the material (a 4th year graph theory module) and solve the exercises without too much difficulty and believe that Graph Theory is something I could be good at. However I remember next to nothing about Number Theory, Group Theory, Rings, Fields, Analysis, Topology, Probability, Set Theory, Logic etc… (basically the whole of the rest of maths). Now at the moment it seems that if one has solid foundation in mathematics, a feel for combinatorics, and the ability to abstract/visualise, this is all you need to do Graph Theory, and that all these other areas are almost entirely separate.
My question is, how long will this go on for? Could I, in theory, get to the frontiers of understanding in Graph Theory and come up with new ideas, with my current level of knowledge about the rest of maths? Can anyone, who is experienced in Graph Theory, provide any insight on this?