Complex sentence: From 1934--91, the two teams played annually as Southwest Conference members (as well as a four-year stretch from 1927-1930), but the matchup ceased in 1991 when Arkansas left the conference to join the Southeastern Conference.
Split into simpler sentences: From 1934--91, the two teams played annually as conference members. The annual matchup ceased in 1991 when Arkansas left the conference to join the Southeastern Conference.