Problem with "or" (conjunction)

The problem is that that the English word "or" does not sufficiently distinguish between inclusive OR and exclusive XOR.

That is why notation is needed in mathematics and logic to address this omission.

You can write briefly:

Let G and H be regular OR planar graphs. (this is the inclusive OR and means that they could be either or both)

Let G and H be regular XOR planar graphs. (This is the exclusive "or" and means they can one or the other but not both.

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However this should really be on the Mathematics site because these terms are not used in ordinary English.

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EDIT

Since OR and XOR are primarily used in logic and your question seems to be about mathematics, it might be better to use set notation. https://www.rapidtables.com/math/symbols/Set_Symbols.html

I don't think the difficulty here arises from the ambiguity between inclusive and exclusive "or". The problem here is that, in general:

Rule 1: "A and B are Z" serves as an abbreviation for "A is Z and B is Z"

but also

Rule 2 "C is X or Y" serves as an abbreviation for "C is X or C is Y."

The sentence in the question has the form "A and B are X or Y", and so we can apply either of the two rules to it (and then apply the other rule to the resulting constituents). The two results so obtained have different meanings, and I see no clear reason to prefer one to the other. The sentence in question is ambiguous, and, unless it's in a context that resolves the ambiguity, it should be rewritten.

Notice that everything I've written here is true if "or" is always interpreted in the inclusive sense. The ambiguity with the exclusive sense is an additional ambiguity, on top of the one that OP asked about.