This is a US centric answer. If you are applying to a mathematics doctoral program in the US and have a mathematics undergraduate degree then the specialty you want to pursue isn't especially important. First, a broad based undergraduate degree has what you need to get started in a US doctoral program. If your undergraduate degree is other than math, you have a more complicated situation, but admission is still possible if you have, in general, "enough" math.
But one reason that this is the case is that such programs start out heavy in coursework, so if you need more combinatorics, for example, you will be able to get it before starting research.
The prerequisites you need for research are probably at the discretion of the advisor you will (later) choose to guide you. They may have some specific suggestions.
The one thing you might need to consider, however, is whether the pure and applied math specialties are contained within a single department or split between departments. Some splitting is common, as having statistics (different example, I know) separate from math. If pure an applied are in a single department, then you don't need to make a selection before applying, and you may be able to delay that decision for a year or two while you take some courses and learn more about possible advisors.
In other countries the situation might be drastically different, of course, with more expected on entry but a shorter time-frame for getting the degree.