How about defining terms, first? If you seriously wish a discussion that has any precision of thought we must all know what is meant by the key terms.
What does exist mean?
Now don't be an idiot and run to the closest dictionary and look up exist=to be, as that means nothing. How about a real, working definition of exist that is not ambiguous? Here is a physics definiton for you: "for something to exist it must have mass and a location in the universe, or it must be observed to have a physically measurable effect on objects that have mass and a location".
You use the word premise, which is correct, in that proof is the hallmark of a system of logic. Unfortunately, our definition of existence cannot be proven by logic. Things cannot be proven to exist. Only ideas can be proven, and they can only be proven to be true or false. We solve true/false propositional dichotomy by a system of logic.
The correct question then is this: what premise does one propose as a starting point in order to prove God a logical necessity?
There is only one possible premise, and it starts lke this: "It is possible that...."
Now, as this is a logic system it follows that any idea can then be placed after the qualifying phrase It is possible: that invisible flying red dragons hover over my bed at night who have the properties of yada, yada, yadi OR we can use Alvin Plantinga as our authority and state It is possible a maximally great being is present in some world who has the properties of yada, yada, yadi.
Both ideas are equally logical. After all, they are only ideas. No idea is more real than another.
We can even construct a logic that will show that invisible flying red dragons are logically real and that maximally great beings are also real - but the reality only concerns itself within the realm of the argument itself. The argument does NOT convert to actuality.
Think me up a ham and cheese sandwich on rye bread, will you? I am sure it is a logical necessity of these rumblings in my stomach.
The best we can ever do with logic is the tuatology that If God exists, God exists. How much more proof than that could anyone want?