The candidates faced a packed week of campaign events and nationally televised debates Monday and Thursday. No Republican has won the party's presidential nomination without carrying South Carolina.
Santorum battled Romney to a virtual tie in Iowa before falling to fifth place in New Hampshire. Gingrich and Perry fared poorly in both states.
All three have the backing of well-financed independent groups known as super political action committees that can help keep their candidacies afloat.
Santorum refused to suggest anyone should drop out of the race as a way to consolidate conservative support behind an anti-Romney candidate. But he said Republicans would have a hard time beating President Barack Obama in November if Romney were the nominee. Santorum cited Romney's push for mandatory insurance coverage in Massachusetts.
Gingrich and Perry used television interviews to focus on Romney's former leadership of the Bain Capital venture capital firm. Both defended raising questions about Bain's business practices, saying Romney's tenure would come under relentless assault from Democrats in the general election.
Former Utah Gov. Jon Huntsman picked up the endorsement of The State, one of South Carolina's leading newspapers. Huntsman came in a weak third in New Hampshire after skipping Iowa, but the paper described him as a "realist" able to appeal to the centrist voters who will decide the general election.
Gingrich, Graham and Scott appeared on NBC's "Meet the Press," while Santorum spoke on "Fox News Sunday" and Perry was interviewed on CNN's "State of the Union."
Associated Press writers Tom Beaumont in Myrtle Beach and Julie Pace in North Charleston contributed to this report.
Follow Beth Fouhy on Twitter at www.twitter.com/bfouhy
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