I make it 9
7 in last years cup run
Auchinleck(H), Saint Johnstone(H), Saint Johnstone(A), Saint Mirren(H), Saint Mirren(A), Celtic(N), Hibs(N)
2 in the league cup
Livingston(H), Dundee United(A)
The best I can find is a run of 12 cup fixtures, spanning 9 ties between 1905 and 1907
Hearts won the cup in 1906 beating Nithsdale Wanderers, Beith, Celtic, Port Glasgow Athletic and Hibs, all at the first time of asking
The Following year Hearts were runners to Celtic up having required replays to dispose of Airdrieonians, Kilmarnock & Raith in the first three ties
Queens Park were beaten in the Semi Final with no need for a replay.