.375 H&H is minimum, but certainly sufficient. There are plenty of other cartridges in this general class (junior medium-bores) that will work, including the 9.3x62 and the .375 Weatherby to name two.
The next step up (senior medium-bores) would include the .404 Jeffery, the .416 Rigby/Remington/Weatherby, and the .458 Win (and a bunch of others). Bigger than that, I probably wouldn't recommend for a first-time Africa hunter who isn't sure what he should be using.
More important than the exact cartridge, by about 100 times, is where the bullet actually goes when you pull the trigger. Only you can do this part correctly. Do it wrong, and you could put yourself or others in danger. A wounded buffalo could trample some poor villager on her way to get water. You need to shoot well, from sticks, in field conditions, and you need to understand the anatomy of the animal. It's amazing to me how many videos I've seen of hunters who invented a reason to try a neck shot, or some other stupid thing at the last minute. Listen to the PH. Also, you need to use a premium bullet and understand its particular strengths and weaknesses. Use a rifle you understand (master). Reliability trumps fancy gizmos and pretty stocks. Also, practice shooting inside of 10 yards. It's amazing how many people screw up a finishing shot from a few feet away.
Did I mention you need to shoot well? I'm talking..... run uphill 100 yards, throw up some sticks, and put every shot in a small paper plate at 50 or 100 yards, every time. Be able to reload on the run, without looking, and figure out how to shoot a long-action rifle without short-stroking the bolt. When the adrenaline is pumping, you're only as good as what you've committed to muscle memory. Some of what typical US hunters have learned transfers to dangerous game, but less than you might imagine.
And then hunt what you want. Only you can figure out that part. Lots of guys go bonkers about warthogs. For others, it's super important to get the various designer color-phases of springbok. For me, I'd rather shoot one truly wild bull than a dozen high-fence cows, but some people are the exact opposite. Also, I'd rather hunt average animals with a great PH than large trophies with a PH who is boring, insufferable, dangerous, or unethical. For others, it doesn't matter as much.