I am a double major in mathematics and electrical engineering. Towards the end of my mathematics major, I began to develop an interest in combinatorics (specifically graph theory), and am seriously considering a Ph.D. in the subject.
Having developed the interest late, my senior thesis will be in a different topic (complex ODEs), but I would have completed one course in combinatorics/graph theory (an upper-level undergraduate course using a grad-level text - I expect a strong reference from the professor teaching this course).
The courses I have completed are basically both the undergraduate and graduate-level algebra sequence, analysis in R and R^n (without a formal introduction to Lebesgue measure), discrete math (for EECS), applied probability, one course in numerical methods, PDEs, applied complex variables, and a "mathematical methods for sciences" course (focusing on special functions, variational calculus, and integral transforms).
I won't have the standard math major courses(i.e, topology, measure theory, and number theory.) I plan to focus more on applications than theory in my combinatorics Ph.D.
Would additional coursework be necessary for a Ph.D. in combinatorics (specifically graph theory)? (I could potentially delay my degree by a year and spend time as a non-degree student at another university, or complete a non-thesis track master's degree before applying for a masters/Ph.d.). Any advice would be appreciated.