Undergraduate prerequisites for a Ph.d. in combinatorics 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.
The prerequisites in combinatorics will not be significantly different from the overall prerequisites in the program to which you are applying.  Having taken a combinatorics course would be beneficial, but there is no need to take many of them as an undergraduate.  You mention Budapest Semesters in Mathematics in a comment.  They offer a lot of combinatorics courses, more than some U.S. universities, and there is certainly no expectation that a typical applicant will have completed this many courses.

In your case, I expect the main issue will be whether you are applying to pure or applied math departments, since combinatorics could be located in either.  For pure mathematics, the admissions committee will wonder how many of your courses were based on rigorous proofs (for example, applied complex variables and mathematical methods for sciences courses might not be), so it would be best to be clear about that.

I would recommend applying broadly and seeing what happens.  Even if your coursework is not ideal, you may still be admitted, with the expectation of having to catch up in a few topics.  Ultimately, it depends on how compelling your application is overall, and your transcript will be just one factor.