The center-right “Alliance” includes the “Centre Party,” which isn’t really center-right. If we take the Centre Party out of the equation, the other members of The Alliance (the Moderate Party, the Christian Democrats and the Liberals—all three are true center-right despite their names) appear to have won (with 6,002 out of 6,004 precincts reporting, so these numbers are pretty much final) a combined 112 seats. Even if those three center-right parties agreed to caucus with the (right-wing, Euroskeptic) Swedien Democrats, the Sweden Democrats appear to have won only 62 seats, so the combined center-right/right seat count would be 174, one seat short of a majority. And since the three center-right parties have said that they would not form a government with the Sweden Democrats, this is all an academic exercise anyways.
The likeliest scenario would be a minority government led by the center-left Social Democrats, in coalition withbthe Green Party and the Left Party, with the Centre Party agreeing to lend them their 31 votes to get to the magic 175 but officially staying in the opposition.
Update: With all votes counted, it appears that the Sweden Democrats won 63, not 62, seats, with the Centre Party (which is a member of the right-of-center “Alliance” but which is not really right-of-center) winning 30 instead of 31 seats.
That means that, if the Moderate Party, the Christian Democrats and the Liberals (i.e., the three true right-of-center members of the Alliance) agree to form a government with the Sweden Democrats, they would combine to have 175 seats, which is exactly the number needed for a bare majority. Of course, while the Sweden Democrats have stated that they would support an Alliance government in return for certain concessions, the Moderate Party, the Christian Democrats and the Liberals all vowed during the campaign that they would not govern in a coalition with the Sweden Democrats, so such result would be unlikely.
But the switch of one projected seat from the Centre Party to the Sweden Democrats does ensure one thing: If the “Red-Green” left-of-center coalition is to form the government, support from the Centre Party legislators would be insufficient to obtain a majority of 175, and they would need the support of at least one member of a right-of-center party. Who knows, maybe they’ll need to call a new election.