The victor will complete the remainder of Giffords' term. Both candidates are promising to run for a full term in the fall, setting up a possible November rematch in a redrawn district that is friendlier to Democrats.

Elsewhere Tuesday, Virginia, Maine, Nevada, Arkansas and South Carolina held primary elections — with most of those states choosing Senate nominees — as did North Dakota, where voters decided to let the University of North Dakota scrap its controversial nickname, the Fighting Sioux.

In Virginia, George Allen brushed aside three conservative Republican rivals in the Virginia primary. Allen's victory set up a November clash with another former Virginia governor, Democrat Tim Kaine, in a campaign closely tied to the presidential race in a state both parties consider vital for victory.

In North Dakota, Rep. Rick Berg defeated businessman Duane Sand in the state's Republican primary.