I'm applying for a PhD to study math with an emphasis on graph theory.
I like graph theory, as well as combinatorics, for a few reasons. First, combinatorics was the first taste that I had of real math. I took a combinatorics class that was way too hard for me while studying abroad in Korea, and I didn't realize I was in over my head until it was too late to drop. With no other options, I asked the TA for help with some binomial identity proofs that I was not at all equipped to do. However, she didn't speak English, so everything she told me was in Korean. I still managed to learn a lot from her and make a huge amount of progress on problems that were previously impossible to me. I eventually did pass the class, even though my grade was very good. The whole of feeling of struggling through math that was too hard for me and resorting a language that I barely knew to get help and still making tangible progress really stuck with me. I felt strongly that no matter how hard things seem, I can learn a lot if I just push through it, and that became my motivation to study more math. I've liked combinatorics ever since.
Combinatorics is closely related to graph theory, which is why I took a graph theory class. I feel that while many branches of math, such as algebra and analysis, really emphasize the rigor of mathematics, graph theory really emphasizes how creative math can be. Any idea can be invented, such as graph colorings, list colorings, perfect graphs, various embeddings of graphs, and random graphs, and there will always be amazing results waiting to be found. I can define a new class of graphs or a new process related to graphs, and I will almost definitely find something new and interesting. Graph theory still has all of the rigor of other types of math, but it's so easy to invent new things, and I feel like the possibilities are endless. I'm sure that other types of math are like this too, but I feel that graph theory has a very inviting and simplistic feel while still being incredibly complex. Graph theory pulls me in with seemingly simple concepts, and then it forces me to attain a deep understanding in order to reach results.
Also, I feel like the ideas of graph coloring and graph structure have a very deep connection that we don't fully understand, and I feel that with enough effort, we'll be able to find the relationship between the way graphs are put together and properties like chromatic number. I want to be a part of this effort.