Well...
sayonara = good bye
ja ne, or just "ja" = see you later (a less formal goodbye, probably more suitable for a boyfriend)
So that's out of the way. Now, boyfriend... "kareshi" or "kare" (pronounced "KAH - reh") is boyfriend, and "sukidoushi" works for boyfriend OR girlfriend, as well as general "soul mate". In other words,
"Ja ne, kareshi." = See you later, boyfriend.
"Ja, kare." = See you later, boyfriend.
"Sayonara, sukidoushi." = Goodbye, soul mate.
Any combination could work. However, I assume most girlfriends would simply refer to their boyfriend by their name and a familiar honorific, such as "-kun".
(Note: I'm not certain on this, but "kareshi-kun" or "kare-kun" might be a fine way to say boyfriend as well, while being more familiar. But don't quote me on that.)
'Relationship' is関係 (kankei) in Japanese.
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