The relationship of company x and company y
The relationship between company x and company y
Which one is the best for the context of law terms?
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I assume you want to designate the relationship between them in order to, for example, describe it (as good, bad, cooperative, competitive, etc.).
The relationship between company x and company y is correct. We normally do not use the other way.
Between is cited here in the example sentence provided for the relevant sense of relationship:
Some English learners may feel that the relationship of x and y is correct because we sometimes use [something] of [something] similarly to 's. For example the entrance of the building, a friend of my grandmother. But we do not generally use this form for a possession of the type X and Y's relationship.