If the problem is that every person on Earth must kill one and only one other person, then there are two ways we could assume this must happen.
One is that you’re not excused from killing just because you’re dead. Even if you die, you must still kill another person in some way–simultaneous bullets, time bombs, and so forth. In this case, everyone on Earth dies, except perhaps for one person if there is an odd number of people on Earth. And even then I think this person must commit suicide.
The other options is that if you’re dead you can no longer kill anyone, so no suicides and no simultaneous or posthumous kills. In this case, I think Oukile must be correct.
Babies and other incapacitated people don’t count in the second scenario. They can’t kill someone, but they can be killed. Everyone incapable of killing another person for whatever reason will end up dead. But this lowers Oukile’s number, which assumes a homogenous population who are all capable of murder and choose a murder victim at random. Hmm. Or does it? Obviously everyone incapable of murder has to be dead at the end of the day, otherwise not everyone will have killed someone. But if everyone has to kill one and only one other person, then what happens if there are large numbers of people incapable of killing, yet every person capable of killing has already killed?
An obvious example, what if there were three people in the world: an able bodied person, a newborn baby, and a coma patient. The able bodied person kills the coma patient. But the baby can’t kill the able-bodied person, and the able person has already killed and can’t kill the baby and so we have a survivor who didn’t kill anyone.